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TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE mm mm.“ The uniqueness of positive solutions to the generalized Lotka-Volterra competition model By Joon Hyuk Kang A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 2000 ABSTRACT The uniqueness of positive solutions to the generalized Lotka-Volterra competition model By J oon Hyuk Kang In this dissertation, we study the uniqueness of the positive solution for the generalized Lotka-Volterra competition model . Some uniqueness results when self-reproduction, self-limitation, and competition rates are positive constants have been obtained over the last decade. We generalize these results when self- reproduction, self-limitation, and competition are more generalized functions. We first establish some generalized uniqueness results when the competition is a gen- eral increasing function. Then some generalized uniqueness results will be obtained when self-limitation and competition are general functions individually. Finally, we also study the case when self-limitation and competition are combined as a more general function. The techniques used in this thesis are elliptic theory, upper-lower solutions, maximum principles and spectrum estimates. The arguments also rely on some detailed properties for the solution of logistic equations. To my family iii ACKNOWLEDGMENTS I would like to express my sincere gratitude to Professor Zhengfang Zhou, my dissertation advisor. All of my work has been done under his patient guidance and many valuable suggestions. I would also like to express my appreciation to my committee members Pro- fessor Dennis Dunninger, Professor Sheldon Newhouse, Professor Michael Frazier and Professor Baisheng Yan for their time and valuable suggestions. Finally I would like to thank my parents, my wife Yunmyung Oh for sharing the difficulties while we are in East Lansing and my close friend Taewan Park for helping typeset. iv TABLE OF CONTENTS 1 Introduction 2 Preliminaries 3 General competition rate 3.1 Uniqueness ............................... 3.2 Uniqueness in a neighborhood of reproduction rates .......... 3.3 Uniqueness in some domain of reproduction rates .......... 4 General self-limitation and competition rates 4. 1 Uniqueness ............................... 4.2 Uniqueness with small perturbation for reproduction rates ..... 4.3 Uniqueness in some domain of reproduction rates .......... 5 Combined self-limitation and competition 5.1 Uniqueness ............................... 5.2 Uniqueness with small perturbation of reproduction rates ...... 5.3 Uniqueness in a region of reproduction rates ............. BIBLIOGRAPHY 16 16 23 33 43 43 50 56 62 63 69 75 82 CHAPTER 1 Introduction For many years, a lot of research has been focused on the Lotka-Volterra compe- tition model r utzAu+u(a—bu—cv) in R+> 0, where f is a strictly decreasing C1 function such that there exists co > 0 such that f (u) S O for u 2 c0. Fiom Lemma 1.1 in [9], we know that the above equation has a unique positive solution when f (0) > A1, where A1 is the first eigenvalue of -A with homogeneous boundary condition. We denote this unique positive solution 38 6f. As a special case f (u) = a - u, the equation Au+u(a—u)=0 in Q, 1439 :O,u> 0 has a unique positive solution when a > A1. We denote this unique solution as wa. We will also use A1(h) to denote the first eigenvalue of —A + h with Dirichlet homogeneous boundary condition. In Chapter 3, we study what happens when we consider the general competition rates. We consider the model Au+u(a—bu—g(v)) =0 in S) (1.2) Av + v(d — fv — h(u)) = o ’ Ulao = ’Ulon = 0, where g, h are Cl, strictly increasing and 9(0) 2 h(0) = 0. We have the following. Theorem 1.1 [fa > A1 + g(%),d > A1 + h(%) and 4bf > [sup(g’(:1:))]2',§R(a,d — h(§)) + 2sup(g’(x))sup(h’(:r)) + [sup(h’(:c))]2%R(d,a — 9(9)), then (1.2) has a f unique coexistence state, where R(A, B) :2: sumeQ fig for A > 0 and B > 0. Theorem 1.2 Suppose a > A1(g(2fd)),d > A1(h(%)). Suppose (1.2) has a unique coexistence state (u,v) and the Frechet derivative of (1.2) at (u,v) is invertible. Then there is a neighborhood V of (a,d) in R2 such that if (a0, do) 6 V, then (1.2) with (a, d) = (a0, do) has a unique coexistence state. Theorem 1.3 Let F be a closed, convex region in R2 such that for all (a, d) E I‘, a > A1(g(“—’f¢)) and d > A1(h(‘=’f)). Suppose that (1.2) has a unique positive solution for all (a, d) E BLI‘ or for all (a, d) E ('33P, where BLI‘ and 831" are the left and right portion of the boundary 0P. Suppose that for all (a, d) E 1", the Frechet derivative of (1.2) at every positive solution to (1.2) is invertible. Then for all (a,d) E F, (1.2) has a unique positive solution. In Chapter 4, we look at the case in which the self limitation and competition rates are general. i.e., consider the model f Au + u(a — 91(u) — 9202)) = 0 (1.3) 4 Av+v(d—h1(v) —h2(u)) :0 in Q, \ uloo = vloo = 0, where 9,-(0) = h,(0) = 0, 9,, h,- are C'1 and strictly increasing for i = 1, 2. We are going to establish the following. Theorem 1.4 Suppose there exist [$1,192 > 0 such that a — A1 — 920:2) > 0,d — 17.1062) < 0,d - A1 — h2(k1) > 0,0. -' 91061) < 0 and 4inf[0,k1](g’1)inf[0,k2](h’l) > a_g 9 — I I sup msupasmsup msupwvrnsupo.)sup(h.>. Then (1.3) has a unique coexistence state. Theorem 1.5 Suppose 91(z) 2 a,h1(z) 2 d for z 2 Co for some constant co > 0. Suppose a > /\1(92(6d_h1)),d > A1(h2(60-g,)). Suppose {1.3) has a unique coexistence state (u,v) and the Frechet derivative of {1.3) at (u,v) is invertible. Then there is a neighborhood V of (a, d) in R2 such that if (a0, do) 6 V, then (1.3) with (a0, do) has a unique coexistence state. Theorem 1.6 Let g1,h1 6 C2, limzxoo 91(2) 2 lim,,_,co h1(z) = 00. Let I‘ be a closed, convex region in R2 such that for all (a,d) E I‘, a > A1(g2(6d_h,)) and d > A1(h2(90_g,)). Suppose that (1.3) has a unique positive solution for all (a, d) E 8LT or for all (a,d) E 33F, where BLI‘ and 03F are the left and right portion of the boundary (9F. Suppose that for all (a, d) E I‘, the Frechet derivative of (1.3) at every positive solution to (1.3) is invertible. Then for all (a,d) E F, (1.3) has a unique positive solution. In Chapter 5, we try to see what happens when we combine the general self limitation and competition rates. We consider the model Au + u(a —— g(u, v)) = 0 (1-4) Av+v(d— h(u,v)) = 0 in 9, “[69 = vloo = 0, where g(0,0) = h(0,0) = O, g,h are Holder continuous up to the second order derivatives in every compact set in R2, all the first partial derivatives are positive. We have the following theorem which generalizes Theorems 1.4, 1.5 and 1.6. Theorem 1.7 Suppose there exist [€1,192 > 0 such that a — A1 — g(0,k2) > 0,d—h(0, k2) < 0,d—A1-h(k1,0) > 0,d—g(k1,0)< 0, and4ian(g§)ian(‘—9—’l) > mpg? M, ",(Su up(5%)) + supi-fi‘Z-Hsuptg-Zl) + 2$111155)sup(f§— 2) where K = [0, k1] X [0,192]. Then (1.4) has a unique coexistence state. Theorem 1.8 Suppose g(z,0) Z a, h(0,z) _>_ d for z 2 co for some constant co > 0. Suppose a > /\1(g(0,0d_h(o,))),d > A1(h(60-g(,0))). Suppose that (1.4) has a unique coexistence state (u, v) and the Frechet derivative of (1.4) at (u, v) is invertible. Then there is a neighborhood V of (a, d) in R2 such that if ((10, do) 6 V, then (1.4) with (a,d) ——— (a0,d0) has a unique coexistence state. Theorem 1.9 Suppose limz_,009(z,0) ——- lim,Hoe h(0,z) : 00. Let I‘ be a closed, convex region in R2 such that for all (a, d) E I‘, a > A1(g(0,0d_h(0,))) and d > A1(h(90_g(,0),0)). Suppose that (1.4) has a unique positive solution for all (a, d) E BLI‘ or for all (a, d) E BRI‘, where BLI‘ and 83F are the left and right portion of the boundary 5T. Suppose that for all (a, d) E F, the Frechet derivative of (1.4) at every positive solution to (1.4) is invertible. Then for all (a,d) E I‘, {1.4) has a unique positive solution. In chapter 2, we will prove some preliminaries to be used later, and some of the statements are well known, and consequently the proofs of them are omitted. CHAPTER 2 Preliminaries In this chapter we will state some preliminary results which will be useful for our later arguments. First, we review the upper-lower solutions to Au+ x,u =0 in $2, (2.1) f( ) ulan = 0 where f 6 Cam x R). Definition 2.1 (1) A function 17. E C 2"'(Q) and satisfying Aa+f(a:,a) g 0 in Q, ’ 11[on 2 0 is called an upper solution to (2.1.) (2) A function u E C2’0(Q) and satisfying Au+ flag.) 2 0 in 9. Elan S 0 is called a lower solution to (2.1}. The main result regarding upper and lower solutions is the following existence theorem. Theorem 2.1 Let f(x,£) 6 00(5) x R) and let u,u_ E C2’O‘(Q) be respectively, upper and lower solutions to (2.1) which satisfy u(x) g u(x),x E {—2. Then (2.1) has a solution u E C2'°‘(Q) with u(x) g u(x) S u(x), x E Q. Secondly, we state some well-known properties of the first(smallest) eigenvalue A101) Of —Au+ xuzAu in $2, (2.2) q( ) ulao = 0, where q(x) is a smooth function from Q to R. Theorem 2.2 (1) The first eigenvalue A1(q) is simple with a positive eigenfunc- tion. (2) If q1(x) < q2(x) for all x E Q, then A1(q1) < A1(q2). (3) ( Variational Characterization of the first eigenvalue) _ . f (IV¢I’+q¢2)dx /\1(Q) — mmoewol(0).d>¢0 fl fn ¢2dx Finally, Gui and Lou [6] proved several properties, which will be useful for our argument, of the unique positive solution wa of the special model Au+u(a-u)=0 in Q, ulan = 0,11 > 0 when a > /\1. Their results can be summarized as the following: Theorem 2.3 (1) 9: is pointwise increasing in 9 when a > A1 is increasing. Especially, wa(x) is an increasing function of a when a > A1. (2) —L¢1[[(;1[[Loo < wa S a, where (bl is the positive eigenfunction with [I $1 ”2: 1 of —A with homogeneous boundary condition corresponding to the first eigenvalue A1. (3) Let m(a) 2 (f9 cog)? Then lima_,)‘1 035% = 7917.1, and#— —> as, in C°° (Q) as a —> A1. (4) For any 52’ compact in 9, we have fQ,(a—wa) S C(Q’, Q), where c is independent of a. In particular, we havega‘l —> 1 almost everywhere in 9 when a -—> 00 Now, we can get the same properties in the generalized model (23) Au + uf(u) = 0 in Q, ”(1'59 2 0,11. > 0, where f is decreasing C1 function such that there exists co > 0 such that f (u) S O for u 2 co. From Lemma 1.1 in [9], we know that the above model has a unique positive solution 6f if f (0) > A1. By the similar proofs in the special model above, 10 we see the following: Theorem 2.4 Iff > 9, then 0, 2 99. PROOF. Note that Agf + 919(6)) < A6f + 6ff(9f) = 0 in Q, Oflaa =0. i.e., 6f is a super solution to Au + ug(u) = O in Q, 1459 = 0,u > 0. Let 0 < e < 1 be small enough such that 669 < 0] in 9. Then A(€69) ‘l‘ 6699(669) : 5(A09 + 0990599)) 2 5(A69 + 099(69» 2 0 in Q, (€69)l69 = 0- Hence, 669 is a sub solution to Au + ug(u) = 0 in Q, ulan = 0. Hence, by the super-sub solution method and the uniqueness of the solution, 09 S 0,. 11 Theorem 2.5 Suppose f > g, f(c0) = g(cl) = 0, and 3’ng 2 23(1). Then 91>92. co-C1 PROOF. It is apparent that u E Cl is a super solution to Au + ug(u) = O in 52, ”log = 0- Let 0 < e < 1 be small enough such that 669 3 c1. Then A(€69) + 6999(569) : €(A‘99 + 699(699» 2 €(A09 + 999(69» 2 O in Q, (€09)[ag = 0. Hence, 669 is a sub solution to Au + ug(u) = O in 52, ”(1'39 = 0,u > 0. Hence, by the super-sub solution method and the uniqueness of the positive solu- tions, 69 3 c1. Since Ag- + 6;%L) = 0 and A2511 + 093%) = O in 9, we have fl_& Co Cl )—0,(9—’ —§)+(o,—og)fl+a(6’ + my) — 99 ”(69% = 0 in a. Co C1 C1 ' Co C 1 A( 12 But, since 6f 2 69 by theorem 2.4, g—gfl 2 9—8?) = 0 and W Z W by assumption, we have Hence, by the Maximum Principle, 2% _>_ 421. Theorem 2. 6 W451 g of 3 c0 L'f—l 5 f’ < 0. PROOF. Let h— — 1911—121- Since co is an obvious super solution, it is enough to show that h is a sub solution of Au+uf(u) = O in Q, Mag 2 0. In fact, Ah+f(h)h = MM +f_(__0)— ¢1f( f(0—) A1 “ML: ll¢ IIL .. II_—L_L¢)II¢>L = (_(__—:1 ll 00 [A¢1+¢Lf([(—%——)1H .. ¢L)l ___ W[-A1¢L+¢Lf([T—-(g)—AH .. ill] 2 “(2)1”L[f(ll(¢)lllL¢l)—/\l]¢l rm)— Z W?” ”(0)" A1)->\1l¢1 in Q, 13 where we used the fact f is decreasing. It is enough to show that J=f(f(0)—)\1)—/\1 20. Since f (c0) = O < A1 3 f (0), we only need to check that for all 0 S x S Co by the Intermediate Value Theorem. But [(0) = 0 and 1’01?) = -[f’(f(0) - f(wl) + llf'($) 2 0. Hence, I(x) Z 0 on [0,c0]. Theorem 2.7 _ 2 l - m f _ 1 Let m(f) — (f9 6f)2. Then llmf(0)_,,\, m9: — fn—¢§($)(limj(0)—u1k(f(1‘)))’ where mm» = fol f’(t6lf(x))dt and ,3?) —+ Ls, in Hgm) LL f(O) —-> A1. PROOF. From AOf +6ff(9f) = 0 in 9, we have —0,A0, — (may) :- o in 9. Hence, by the Green’s identity and the fact f (6 f) S f (0), AMA? = [Damn 14 Mug ‘8‘.” (9/) (0) f(0)/99 s f(0)/99§ = f(0)[m(f)]2. which implies that V9f |2< L'TII) f(0) (b weakly as f (O) ——> A1. Further- more, d satisfies llng]2 < A1,] (152: f(lim 6—1—— )2 =1 0M1 n m(f) by the Sobolev Imbedding Theorem. Therefore, (25: d1, and——‘L —> p1 strongly m(floll in H662) and then in H362) as f(O) —+ /\1 by the standard elliptic regularity theory. Moreover, we have 9 f(0) ¢1Amm +21%!) :0 mInA‘l’1 + ’\1mII>¢1 Z 0 in 9. Hence, by the Green’s second identity, (f(Ol—Arlfna51(;6(f73) = [WI (f(0)(—f )f9( >> m(f)/(b 0f(f )[_0) f(gfll [m(f >12 ' Hence, But, Hence, 15 m(f) Z fn¢17ngfi f(0) - A1 lo ¢1°{91/m(f)}'{(f(0)- “Ml/m(f)}. f(O) — MILL) = [01%f(t9;(w))dt = 0 f'IwLIvWLIL) m(f) I IL¢LIL>I%I.§%,’> f (0) - *1 IL ¢L(z)%§j§ ““5332” —> 1 as f(O) _> A1) fa —¢‘i($)(1imfI0)—>.\L k(f (110)) where k( f (x)) is as in the theorem. Theorem 2.8 For any (2’ compact in 9, we have f9,(f(6f)) S c62’,S2), where c is independent of f. PROOF. From A9; + 9ff(6f) = O and I9, 2 0, we know that . fa lV¢l2 f —— = 1. ¢€lll§(fl) fa f(6';)<152 Then by choosing a function (150 6 H362) such that ¢o(x) 2 co > O in 12’, we obtain 1 2 _ , [new 3 — l. IvLOI _ cm ,9, CHAPTER 3 General competition rate In this chapter, we consider the Lotka - Volterra model with general competition rate. To be precise, r Au+u(a—bu—g(v))=0 in Q, (3.1) l Av + v(d —- fv — h(u)) = 0 \ uloo = vlon = 0L where g,h are Cl, strictly increasing and 9(0) 2 h(O) = 0, for example, we can consider g(x) 2: h(x) = 1n(x + 1). We try to generalize the uniqueness result of Cosner and Lazer [3] to this model. We also consider the effect of perturbation of reproduction rate. 3. 1 Uniqueness In 1984, Cosner and Lazer [3] proved that if the self reproduction rates a, d are large and the self limitation rates b, f are large, i.e, the competition rates c, e are 16 17 relatively small, then the Lotka-Volterra competition model Au+u(a—bu—cv)=0 inS2, l Av+v(d—fv—eu)=0 L Uloo = ’Ulao = 0 has a unique coexistence state. The precise statement is the following: Theorem 3.1 If a > A1 + $2 > A1 + g and 4bf > Lg‘iRIa, _ “78) + 2ce + “72R(d,a — %), then the above model has a unique coexistence state, where R(A, B) = 31113er m for A > O, B > 0. £08 (13) The following is our main result in this section. Theorem 3.2 Ifa > /\1 + g(%),d > A1 + h(%) and 4bf > [sup(g’(x))]2£R(a,d —— h(%)) + 2sup(g’(x))sup(h’(x)) + [sup(h’(x))]2%R(d,a - g(Sjl-)), then (3.1) has a unique coexistence state, where R(A, B) 2 supra, “(3 for A > O and B > 0. W8 PROOF. By the Maximum Principle, %wd But, since 9 is increasing, d < 7. 960%) S 56%), and SO AKA + (a - 96%))” Z A1(A + (a — g(%))1) > 0, since a ~— g(%) > /\1 > 0. Similarly, we have A1(A + (d — h(-,1;wa))I) > 0. Hence, by The- orem 1.1 in [9], (3.1) has a positive solution. We concentrate on the uniqueness part. Suppose (u, v) is a positive solution to (3.1). Then Au+u(a — bu) = ug(v) > 0 in 9) ulan = 0. 18 Hence, u is a sub solution to Az+z(a—bz)=0 inS2, Zlag = 0. Any sufficiently large constant is a super solution to Az+z(a—bz)=0 inS2, Zlag = 0. Hence, by the super-sub solution method, we have (3.2) u 3 if. The same argument shows (3.3) v S 9f“. Since wd < d and g is increasing, g(v) S g(l’fi) S g(%) and we have Au+u(a—g(%)—bu)_<_Au+u(a-bu—g(v))=0 inS2, Ulan = 0L which means that u is a super solution to Az+z(a—g(-‘-}-) —-b2) 2 0 in 9, Zlon = 0- 19 Let 451 be the first eigenfunction of Au+)\u=0 inS2, 1439 = 0. If e > 0 is so small that a — g(%) — A1 — ebol > 0 on $2, then d d A6951 + €¢1(a — 9(7) — 126051) : 6[A9751 + (151(0 — g(fl — b€¢1ll > €(A¢1 + 31491) = O inQ, which implies that eqbl is a sub solution to Az+z(a—g( )—bz)=0 1119, i f 2'39 2 0. Hence, by the super-sub solution method, 1 (3-4) 3wa_g(%) g u. The same argument shows (3'5) %wd_h(%) S ’U. 20 From (3.2) to (3.5), 1 Ewe-LI?) u , w _ 2 ’U . (3.6) f Consequently, for any positive solution (u, v) of (3.1), the inequality (3.6) hold. Now we are ready to prove the uniqueness. Suppose (u1,v1) and (u2,v2) are positive solutions to (3.1). Let p 2 ul — ug and q 2 v1 —— v2. Then Ap + (a — bul — g(U1))P = Aul — Aug + (a — bul — g(v1))(u1 — U2) = —AU2 — (a — bul — g(vlllu2 = —Au2 — (a — bug — g(vg) + (711.2 - bul +9012) - g(v1))U2 g(v2) g(U1)) in Q b b = —bu2(u2 — 111 + Hence, (3-7) AP + (a — bul — g(vl))p — b11213 — u2(9(’01)— 9012)) = 0 in Q- The same argument shows that (3.8) Aq + (d — f’U2 — h(u2))q — fvlq — v1(h(u1) — h(u2)) = 0 in S2. Since A1(A + (a — bul — g(v1))I) = O, by the Variational Characterization of the 21 first eigenvalue, if z 6 C262) and ZlaQ = 0, then (3.9) f92(—Az — (a — bul — g(v1))z)dx Z O. The same argument shows that if w 6 C262) and ”Man = 0, then (3.10) fnw(—Aw — (d — f’Uz —- h(u2))w)dx 2 O. From (3.7) and (3.8), we have -pAp - (a - b’uL — g(vi))ID2 + bier? + U2p(g(v1) - g(v2)) = 0. —qu — (d — f’Ug — h(ug))q2 + fv1q2 + v1q(h(u1) — h(u2)) = 0. Hence, from (3.9) and (3.10), we have fnlb'uu?2 + U2p(g(v1) - 9022)) + v1q(h(ul) —- h(u2)) + fvlqzld-T s 0. Hence, by the Mean Value Theorem, for each x 6 S2, there exist «5, ii such that 9(01) — 51(02): 9'(’5)(’01 - v2) 2 9,61qu h(UI) — h(u2) = ”(WWI — “2) = Wall)- Hence, fIbuLzr + IuLg'Iv + th'Ivnoq + vaqudx 3 0. Hence, p E q E 0 if bugC2 + (ugg’(v) + vlh’(u))Cr] + fv1n2 is positive definite 22 for each x 6 S2, which says ugg’w)2 + vfh’(u)2 + 2u2v1g’(v)h’(u) — 4bfu2v1 < 0 for each x €— 52. i.e., (3.11) 4b f > 159-9117)? + 2g'(o)h'(a) + ;U—l-h’(u)2for each L e a. 1 U2 But, from the inequality (3.6), 491W + zg'IthIv + em)? wi— g,g'ar + 2g'(v )h (u )+ r} i,h’Iu > (312) = 9’(I7)2£R(aLd-h(%))+ 2g’v()h’(u)+h’(u)23R(da— LI?» [sup(g’( A1(g(£fd)), d > A1(h(-“f)). Suppose (3.13) has a unique coexistence state (u,v) and the Frechet derivative of (3.13) at (u,v) is invertible. Then there is a neighborhood V of(a, d) in R2 such that if(a0, do) 6 V, then (3.13) with (a, d) = (a0, do) has a unique coexistence state. PROOF. Since the Frechet derivative of (3.13) at (u, v) is invertible, by the Im- plicit Fimction Theorem, there is a neighborhood V of (a, d) in R2 and a neigh- borhood W of (u, v) in [Cg‘L‘J'6—2)]2 such that for all (a0, do) 6 V, there is a unique positive solution (uo, v0) 6 W of (3.13). Suppose the conclusion of the theorem is false. Then there are sequences (an,dn,un,vn), (an,d,,,u,’,,v;) in V x [C'g‘LO’6-2)]2 such that (u,,,v,,) and (u‘ v") are the positive solutions with (a,d) = (amdn) n, n and (umvn) 51$ (u‘ v“) and (amdn) —> (a,d). By the standard elliptic theory, 711 fl. 24 (umvn) —> (u,v), (u;,v,‘;) —+ (u*,v*). Phrthermore, (u,v) and (u*,v*) are solu- tions of (3.13) with (a,d). Claim 11 > 0, v > 0,u“ > 0, v* > 0. It is enough to show that ii and v are not identically zero, because of the Maximum Principle. Suppose not, by the Maximum Principle again, one of the following cases should hold. (1) a is identically zero and i7 is not identically zero. (2) ii is not identically zero and v is identically zero. (3) a is identically zero and v is identically zero. (1) Suppose a is identically zero and v is not identically zero. Let ii}, = “THIS-3’5" : vn for all n E N. Then Au}, + iin(a,, — bun — g(v‘n)) = 0 in S2. A + v~n(dn — fv}, —— h(un)) = 0 Using elliptic theory, tin -—> it in 02’“ ,and A2] + u(a — 9(6)) 2 0 . in S2 Av+v(d—fv) =0 since g,h are continuous and h(O) = 0. Hence, a = A1(g(v)) and v = ‘ifd. Hence, a 2: A1(g(gfd)), which is a contradiction to our assumption. (2) Suppose u is not identically zero and ’U is identically zero. Let u“,, 2 un and v}, = Hill: for all n E N. Then A6,, + ii,,(a,, — bu}, — g(vn)) = O in 12. Av}, + ii'n(d,, — fvn — h(iL,)) = O 25 Using elliptic theory again, 1],, —> i) in 02'“ ,and Au+u(a—bu) =0 in (2 A1? + v(d — 11(21): 0 since g, h are continuous and g(0) = 0. Hence, a = 9’3“ and d = A1(h(a)). Hence, d = A1(h(E’,-f)), which is a contradiction to our assumption. (3) Suppose both ii and i} are identically zero. Let 11:, = ——“L and v), = —3n— for allnE N. Then llunlloo llvnlloo Au}, + ii',,(a,, — bun — g(vn)) = 0 in S2. A6,, + 17,,(dn — fvn — h(un)) = 0 Then u",, —+ a and v”,, —> v by elliptic theory, and A2] + ail = 0 in (2 A1”) + do = 0 since g, h are continuous and g(0) = h(0) = 0. Hence, a = d = /\1, which is a contradiction to our assumption. Consequently, (a, 17) and (u‘, v") are coexistence states with reproduction rates (a, d). But, since the coexistence state in this case is unique by assumption, (a, v) = (u‘, v“) = (u, v), which contradicts the Implicit Function Theorem. 26 Corollary 3.4 Consider the original Lotka - Volterra model Au+u(a—bu—cv)=0 inQ, Av+v(d—fv—eu)=0 Uloo = vlea = 0L where a,c,d,e are constants such that a > A164?) and d > A1643“). Suppose the above model has a unique coexistence state (u, v) and the Frechet derivative of the above model at (u,v) is invertible. Then there is a neighborhood V of (a,d) in R2 such that if (ao,d0) E V, then the above model with (a,d) = (a0,d0) has a unique coexistence state. PROOF. It is an immediate consequence of Theorem 3.3 by choosing g(v) = cv, h(u) = eu. Corollary 3.5 ( Theorem 3.2 in [2]) Consider the model , Au+u(a—u—cv)=0 inQ, l Av+v(d—v—eu)=0 [ Uloo = vloo = 0L where a,c,d,e are constants such that a > A1 and 0 < c,e < 1. Then there is a neighborhood V of (a, a) in R2 such that if (a0,d0) E V, the above model with (a, d) = (a0, do) has a unique coexistence state. PROOF. From [3], if a = d and 0 < c,e < 1, then the above model has a unique coexistence state (u, v). Furthermore, from [2], we know that if a = d and 27 0 < c, e < 1, then the Frechet derivative of the above model at (u, v) is invertible. Hence, Corollary 3.4 implies the desired result. In 1991, Lopez-Gomez and Rosa Pardo proved a uniqueness result by the fol- lowing theorem in [7]. Theorem 3.6 Consider the model [ Au+u(a—u—cv)=0. 4 m9, (3-14) Av + v(d — v — eu) = 0 1 Uloa = ’Ulon = 0L where a,c,d,e are constants such that a > A1(cwd),d > A1(ewa) and 0 < c,e < 1. Suppose supn msupn m S i, where if) = wa(cwd) is the unique positive solution of the equation (—A + cwd)w = aib — 1b? in S2, 1,!) = 0 on (99 and similarly for ibd(ewa). Then {3.14) has a unique coexistence state. But, we can find that the uniqueness is guaranteed not only for one point (a,d) but for a neighborhood of (a, d). Corollary 3.7 In theorem 3.6, there is a neighborhood V of (a, d) such that if (ao,do) E V, then (3.14) with (a,d) = (a0,d0) has a unique coexistence state. PROOF. Theorem 3.6 says (3.14) has a unique coexistence (u, v). But from [7], the condition in Theorem 2.4 guarantees the invertibility of the Frechet derivative of (3.14) at (u,v). Hence, it follows from Corollary 3.4. 28 In 1989, Cantrell and Cosner obtained a uniqueness result in a region by the following theorems in [2]. Theorem 3.8 Consider the model f Au+u(a—u—cv)=0 (3.15) ( Av+v(d—v—eu) =0 in Q, 1 Ulon = vloo = 0 where a,c,d,e are positive constants and 0 < c,e < 1. Let /\1 < do < ao 1 3c—i2ce—cze < —ce and BMW] = mifldogagoo ||(—A+2wL—a)-1||Hwall' Suppose do > A1(ewao), BMW} and e(3Ko —- 1) < Bldom]: where Ko = supdoSdSaSao 3:. Let VMMO] = {(a,d) : do S d S a S ao}. Then for all (a, d) E V[do,oo]L (3.15) has a unique coexistence state. Theorem 3.9 We consider the same model (3.15) in Theorem 3.8. Let A1 < do < ao and do > A1(ewao). Suppose W31 > (cKo + e)2, where Ko = supdoSdSaSao 5:. Let V[do,ao] = {(a,d) : do 3 d S a S ao}. Then for all (a,d) E l/[dmaO], (3.15) has a unique coexistence state. But, we can find that the uniqueness is guaranteed not only for the closed region mentioned above(V[do,aol) but for an open set containing l/[doflo]. We will see later that Theorems 3.8 and 3.9 are special cases of our Theorem 3.15. Corollary 3.10 We consider the same model (3.15) in Theorem 3.8 and consider the same conditions in Theorem 3.8 or 3.9. Then there is an open set V containing l/[dmao] such that if (a, d) E V, (3.15) has a unique coexistence state. 29 PROOF. From the results of Theorems 3.8 and 3.9, we know that for all (a,d) E Vole“), (3.15) has a unique coexistence state. Furthermore, from [2], the invertibility of the Frechet derivative of (3.15) at the unique solution (u, v) for each (a, d) E 14,10,001 is guaranteed. Hence, by Corollary 3.4, for each (a, d) E VMMO], there is an open neighborhood WW1) of (a,d) in R2 such that if (a',d) E Wow), (3.15) has a unique coexistence state for every (a, d) E U(0.d)€VI.LO,LO) W(a.d)° From the argument above, it is important to get some condition to guarantee the invertibility of the Frechet derivative of the generalized model (3.13). Theorem 3.11 Consider the model Au+u(a—bu—g(v)) =0 inf2 (3.16) Av+v(d— fv—h(u)) = 0 ulan 2 ”(flag 2 0. Suppose (u, v) is a positive solution to (3.16). If4bfuv > [(sup(g’))u+(sup(h’))v]2, then the Frechet derivative of (3.16) at (u, v) is invertible. PROOF. The Frechet derivative of (3.16) at (u, v) is —A + 2bu + g(v) — a ug’(v) vh’(u) —A+2fv+h(u)—d 30 By Fredholm Alternative, we need to show that the solution of -Ar + (21m + g(v) - (1)99 + g’(v)m/) = 0. -A1/J + h’(u)v

[(sup(g’))u + (sup(h’))v]2 implies that the integrand in the left hand side is positive definite in {2 which means that 90 = If) E 0, i.e., the invertibility of Frechet derivative. Corollary 3.12 ([2]) Consider the model Au+u(a—u—cv) =0 (3.17) in S2. Av+v(d—v—eu)=0 31 Suppose (u,v) is a positive solution to (3.17) and 4uv > (cu + ev)2. Then the Frechet derivative of (3.17) at (u, v) is invertible. PROOF. It follows immediately from Theorem 3.11. Now, we try to get other uniqueness results. We continue to consider the same model I Au + u(a — bu — g(v)) = 0 in 52. (3.18) 4 Av + v(d — fv — h(u)) = 0 1 Uloa = vloo = 0 Then we can derive the following uniqueness results. Corollary 3.13 Suppose a > A1 + g(%),d > A1 + h(%) and 4bf > [sup(g’) + sup(h’)? sup mflsupw'fi sup “dim +sup(h’)], then there is a neighborhood V of (a, d) in R2 such that if (ao,do) E V, then (3.18) with (a,d) = (ao,do) has a unique coexistence state. PROOF. From wa < a,cud < d, we know that a > A1(g(‘-"f)) and d > A1(h(“—’gl)). b (dd f wa 4bf > [sup(g’) + sup(h)? sup w d llsumg’); 8111) cu W) + sup(h’)l a-9(7) ’ 3 f wL = glsup(g’)l2sur> + sup(g’)sur>(h’) wd-h(%) L , b , . + sup w sup wd sup(g') sup(h ) + —[sup(h )]2 sup war—Mt) “IL—LI?) f Iva—LI?) f we 2 glsup(g’)l281m +sup(g’)suplh’) wd—h(%) can (.1) b w + d sup(g')sup(h’) + -lsup(h’)l2sur> d “cl-Mt) “42-ng f “Jo—LI?) 32 b +2supIg'>supIh')+—IsupIh'>12sup “’0' , “dd-Mt) f ”LL—LI?) we 2 98111300}? sup Since wa > w a_g(%),wd > wd_h(%). Therefore, by Theorem 3.2, (3.18) has a unique coexistence state (u, v). Furthermore, by the estimation of the solution in the proof of Theorem 3.2, I I b w I (.00 I 4bf > [sup(g)+sup(h)7supw——:;f—)l[sup(g)%supm+sup(h)l a—g — a 2 [sup(g’) + sup(h')§IIsupIg')§ + sup(h’)l- Hence, 4bfuv > [sup(g')u + sup(h’)v]2 Hence, by Theorem 3.11, the Frechet derivative of (3.18) at (u, v) is invertible. Therefore, the conclusion follows from Corollary 3.4. Corollary 3.14 Consider the model Au+u(a—bu—cv)=0 inf2, Av+v(d—fv—eu)=0 ’ulaQ = 0'39 2 0. Suppose a > A1+%,d > A1+af and 4bf > [c+-bj+E sup fiflgf sup fi+sup(hl)], then there is a neighborhood V of (a, d) in R2 such that if (ao,do) E V, then the above model with (a, d) = (ao, do) has a unique coexistence state. 33 3.3 Uniqueness in some domain of reproduction rates Consider the model I Au+u(a—bu—g(v))=0 . l 1n {2, (3-19) Av + v(d — fL — h(u)) = 0 K Uloa = Uloa = 0. In this section, we try to find a region of reproduction rates a and d that guarantees the existence of a unique positive solution to (3.19). The following is the main theorem. Theorem 3.15 Let F be a closed, convex region in R2 such that for all (a, d) E F, a > A1(g(£fd)) and d > A1(h(9,-f)). Let BLF = {(Ad,d) E FlFor any fixed d, Ad = inf{a](a,d) E F}}. Suppose that (3.19) has a unique positive solution for all (a, d) E BLI‘. Suppose that for all (a, d) E I‘, the Frechet derivative of (3.19) at every positive solution to (3.19) is invertible. Then (3.19) has a unique positive solution for all (a, d) E F. PROOF. For each fixed (1, let Ad = sup{a : (a, d) E F}. We need to show that for every a such that Ad 5 a 3 Ad, (3.19) has a unique positive solution. Since (3.19) with (a, d) = (Ad,d) has a unique positive solution (u,v) and the Frechet derivative of (3.19) at (u, v) is invertible, Theorem 3.3 implies that there is an open neighborhood V of (Ad,d) in R2 such that if (ao, do) E V, then (3.19) with (a, d) = (ao, do) has a unique positive solution. 34 Let A, = sup{Ad S A S Ad : (3.19) has a unique coexistence state for Ad S a S A}. We need‘to show that A, = Ad. Suppose A, < Ad. From the definition of As, there is a sequence {An} such that An —> A; and there is a sequence (un, vn) of the unique positive solution of (3.19) with (a, d) = (An, d). Then by the Elliptic theory, there is (no, vo) such that (un, vn) converges to (no, vo) uniformly and (uo, vo) is a solution to (3.19) with (a, d) = (A,, d). We claim that no is not identically zero and vo is not identically zero. Suppose this is false. Then by the Maximum Principle, one of the following cases should hold: (1) uo is identically zero and vo is not identically zero. (2) uo is not identically zero and vo is identically zero. (3) Both no and vo are identically zero. (1) Suppose no is identically zero and vo is not identically zero. Let if": TI_L— and v",, =v,, for allnE N. Then u unlloo Allin + dn()‘n — bun — 90611)) = 0 in (2, Av}, + v~n(d —— fa, — h(u,,)) = 0 and ii}, ——> a uniformly in S2 by elliptic theory, and Ail + u(A, —- g(vo)) = 0 . 1n S2 A’Uo + ’Uo(d — f’UO) = 0 since 9, h are continuous and h(0) = 0,which implies that vo = 9f“ and A, = A1(g(vo)) = A1(g(3f4)), a contradiction. (2) Suppose uo is not identically zero and vo is identically zero. The same argument leads to uo = 33‘- and d = A1(h(uo)) = A1(h(“’—;-‘L)), a contra- 35 diction to the hypothesis. (3) Suppose uo = vo = 0. Let ’11:" = ”—131;- and 17": “it“: for allnE N. Then Au), + a..(,\,, — bun — g(vn)) = 0 in 9 Av}, + v”,,(d — fvn — h(un)) = 0. tin ——> a and v2, —> v uniformly by elliptic theory, and Au+A3u=0 inf2 Aii+dv=0 since 9, h are continuous and 9(0) = h(0) = 0, which implies that d = A, = A1, which is impossible. Consequently, uo is not identically zero and vo is not identically zero. We claim that (3.19) has a unique coexistence state with (a, d) = (A,,d). In fact, if not, assume that (u1, v1) ¢ (uo, vo) is another coexistence state. By Implicit Function Theorem, there exists a Ad < ii < A, and very close to As, (3.19) has a coexistence state very close to (u1,v1) which means that (3.19) has more than one coexistence state for (a, d) = (a, d). This is a contradiction to the definition of A8. But since (3.19) has a unique coexistence state with (a, d) = (A,,d) and Frechet derivative is invertible, Theorem 3.3 concluded that A, can not be as defined. By the similar argument, we can prove the following: 36 Theorem 3.16 Let F be a closed, convex region in R2 such that for all (a, d) E F, a > A1(g(‘-"}4)) and d > A1(h(5“)). Let 83F = {(Ad,d) E FIFor any fixed d,Ad = sup{a|(a,d) E I‘}}. Suppose that (3.19) has a unique positive solution for all (a,d) E 63F. Suppose that for all (a,d) E I‘, the Frechet derivative of (3.19) at every positive solution to (3.19) is invertible. Then for all (a,d) E F, (3.19) has a unique positive solution. PROOF. For each fixed d, let Ad = inf {a : (a,d) E I‘}. By the similar argument in Theorem 3.15, we can show that for every a such that Ad S a S Ad, (3.19) with (a, d) has a unique positive solution, and so the theorem follows. We can see that Theorem 3.8 and Theorem 3.9 are results from the Theorem 3.15 and the Theorem 3.16. Corollary 3.17 (Theorem 4.5 in [2]) Consider the model I Au+u(a—u—cv)=0 infl, (3.20) l Av+v(d—v—eu) =0 1 uloo = ’Uloa = 0 where a,c,d,e are positive constants and 0 < c,e < 1. Let A1 < do < ao 1 1 e< —ce _A+2wa_a)_1““%“. Suppose do > A1(ewao), and Bldo.ao] : mindoSaSao ||( BMW“); and e(3Ko — 1) < Buom], where Ko = SUPdostasao 5:. Let V[d0,ao] : {(a, d) : do S d S a S ao}. Then for all (a, d) E V[dO,001, (3.20) has a unique coexistence state. 37 PROOF. Since 0 < c,e < 1, by [3], (3.20) has a unique coexistence state for all (a, d) E R2 such that do S d = a S ao. Furthermore, the above conditions guarantee the invertibility of the Frechet derivative of (3.20) for all (a, d) E V[do.ao] by [2]. Hence, the corollary follows from Theorem 3.15. Corollary 3.18 (in [2]) We consider the same model (3.20) in Corollary 3.17. Let A1 < do < ao and do > A1(ewao). Suppose W > (cKo + 6)“), where Ko = supdoSdSaSao 5:. Let V[do.ao] = {(a,d) : do S d S a S ao}. Then for all a,d E V a , 3.20 has a unique coexistence state. ldo, o] PROOF. Since 0 < c, e < 1, it is proved in [3] that (3.20) has a unique coexistence state for all (a,d) E R2 such that do S d = a S ao. Furthermore, the above conditions guarantee the invertibility of the Frechet derivative of (3.20) for all a, d E V a by 2 . Hence, the corollary follows from Theorem 3.15. [d0, 0] Using Theorem 3.16, we can also get the similar results as above. Corollary 3.19 Consider the model (3.20). Let A1 < ao < do and Blao.do] = ' 1 3e—2c —e2c mmaoSano II(—A+2wd—d)‘1llllwdll‘ Suppose ao > A1(cwd0),-—£—— < BEND] and l—ce C(3K0 — 1) < Blamdo]! where K0 = supaoSanSdo fig. Let WOOLdol = {(a,d) I 620 < a S d S do}. Then for all (a,d) E l/[amdo], (3.20) has a unique coexistence state. Corollary 3.20 We consider the same model (3.20) in Corollary 3.17. Let A1 < ao < do and ao > A1(cwdo). Suppose W92 > (eKo + c)2, where Ko = supaoSanSdo 5:. Let V[a0,do] = (a, d) : ao S a S d S do. Then for all a,d E V d , 3.20 has a unique coexistence state. [0'01 Ol 38 Actually, if we consider the result in section 2, then we can extend the region F in the Theorem 3.15 and the Theorem 3.16 to an open set including F to guarantee the unique coexistence state. Corollary 3.21 Let F be a closed, convex region in R2 such that for all (a, d) E F, a > A1(g(5fd)) and d > A1(h(5b“)). Let BLF = {(Ad,d) E FIFor any fixed d,Ad = inf{a|(a,d) E F}}. Suppose that (3.19) has a unique positive solution for all (a,d) E BLF. Suppose that for all (a,d) E F, the Frechet derivative of (3.19) at every positive solution to (3.19) is invertible. Then there is an open set W in R2 such that F C W and for every (a, d) E W, (3.19) has a unique coexistence state. PROOF. From the results in Theorem 3.15, for each (a, d) E F, (3.19) has a unique coexistence state (u,v). Furthermore, by the assumption, for each (a, d) E F, the Frechet derivative of (3.19) at the unique solution (u, v) is invertible. Hence, Theorem 3.3 concludes that there is an open neighborhood V(a.ci) of (a, d) in R2 such that if (ao, do) E V(a.d)1 then (3.19) with (a, d) = (ao,do) has a unique coexistence state. Let W = U(a’d)e1‘ v(a.d)- Then W is an open set in R2 such that F Q W and for each (ao, do) E W, (3.19) has a unique coexistence state. Similarly, we can start from the right half of the boundary. Corollary 3.22 Let F be a closed, convex region in R2 such that for all (a, d) E F, a > A1(g(5fd)) and d > A1(h(5b“)). Let 63F = {(Ad,d) E FIFor any fixed d, Ad = sup{a|(a,d) E F}}. Suppose that (3.19) has a unique positive solution for all (a, d) E 63F. Suppose that for all (a,d) E F, the Frechet derivative of (3.19) at 39 every positive solution to (3.19) is invertible. Then there is an open set W in R2 such that F C W and for every (a, d) E W, (3.19) with (a,d) has a unique coexistence state. At this point, we can see that Corollaries 3.21 and 3.22 are extensions of Theorem 3.3, and so Corollaries 3.4, 3.5, 3.7, 3.10, 3.13 and 3.14 are special cases of Corollary 3.21 or Corollary 3.22. Using these theories, we can obtain more uniqueness result. Corollary 3.23 Suppose (ao, d) E R2 is such that ao > A1 + 9(5), (1 > A1 + h(5), where ao < M. Suppose 2 311p llsup(g’)i sup + sup(h’)l- 4b ’ h’ f > [sup(g ) + sup( )f woo—1K?) b wd_,,(131) Then for every (a, d) such that ao S a S M, (3.19) has a unique coexistence state. Furthermore, there is an open set W in R2 such that {(a.d) : ao S a S M} C W and for all (a, d) E W, (3.19) has a unique coexistence state. PROOF. From wa < a, wd < d, we know that for all a such that ao S a S M, a 2 ao > A1(g(-“—}¢)) and d > A1(h(5b“)). 4bf > IsupIg'>+sup(h'19sup—5‘i—1IsupIg'1isup——“’M—+sup(h'>1 f woo—gId/n b wd—hIM/b) f 01M = 3 [sup(g’fl2 811p + sup(g’) sup(h’) “Li—Ml?) w w b + sup ——L— sup ———d—— sup(g’) sup(h’) + —[sup(h')]2 sup wd—hIM/b) woo—gId/n f “ho—9g) 40 f wa L 2 glsup(g’)128up ° +sup(g')sup(h) LI’d—h(igl) wa w b w + ° d sup(g’)sup(h’)+-[sup(h’)l281m d cud—nag) woo-16%) f ”GO—9"?) wa b w 2 §IsupIgiIqup ° +28up(g’)sup(h’)+-[sup(h’)l2sup d LUd_h(‘_‘gl) f aO—g(%l since woo > wao_g(%),wd > wd—Mglll' Therefore,(3.19) with (ao, d) has a unique co- existence state from Theorem 3.2. Furthermore, using the estimate of the solution in the proof of Theorem 3.2, for all a such that ao S a S M, 4bf > [SUp(g’)+sup(h’)Esup wd llSUp(g')£sup wM +sup(h')l f ”co-9%) b wot—h(Ag) I I b I f can I > [sup(g)+sup(h)-Sup llsup(g)-sup +sup(h)l f wa—g(%) b “id—M95) I I v I u I 2 [sup(g ) + sup(h )Ellsupw ); + sup(h )l, where (u, v) is any positive solution to (3.19) with (a, d). Hence, 4bfuv > [sup(g’)u + sup(h')v]2. Therefore, Theorem 3.11 says for all a such that ao S a S M, the Frechet derivative of (3.19) at any positive solution (u, v) to (3.19) is invertible. Hence, the theorem follows from Corollary 3.21. By the similar argument, we can also get the following: Corollary 3.24 Suppose (ao, d) E R2 is such that M > A1+ g(5), d > A1 + h(flb‘l), 41 where ao > M. Suppose Mao + sup(h’)]. 4bf > [sup(g’) + sup(h); sup IIsupIgi§ sup CUM-fl?) “id—h(flbfl) Then for every (a, d) such that M S a S ao, (3.19) has a unique coexistence state. Furthermore, there is an open set W in R2 such that {(a, d) : M S a S ao} C W and for all (a, d) E W, (3.19) has a unique coexistence state. Now, we consider the specific model Au+u(a—bu—cv) =0 in {2, (3.21) Av + v(d — fv — eu) = 0 “loo = Uloa = 0 where a,c,d,e are positive constants. From Corollaries 3.23 and 3.24, we can get the following uniqueness results regarding (3.21). Corollary 3.25 Suppose (ao,d) e R2 is such that L1,, > A1 + 764ml > A1 + iii where ao < M. Suppose 4bf > [c+ 5 sup WW); sup if; + e]. Then for every (a, d) such that ao S a S M, (3.21) has a unique coexistence state. Furthermore, there is an open set W in R2 such that {(a,d) : ao S a S M} C W and for all (a, d) E W, (3.21) has a unique coexistence state. Corollary 3.26 Suppose (ao,d) E R2 is such that M > A1 + 5h > A1 + 9?, where ao > M. Suppose 4bf > [c + 5 sup 554;“? sup 53:5— + e]. Then for every M- d— 42 (a, (1) such that M S a S ao, (3.21) has a unique coexistence state. Furthermore, there is an open set W in R2 such that {(a,d) : .M S a S ao} C W and for all (a, d) E W, (3.21) has a unique coexistence state. CHAPTER 4 General self-limitation and competition rates In chapter 3, we obtained some uniqueness result for general competition rates. In this chapter, we try to get uniqueness result in more generalized model, that is, the model with general self-limitation and competition rates. We consider A“ + u(a — 91W) “ 92(9)) 2 0 in Q, (4.1) Av + v(d — h1(v) -— h2(u)) = 0 A 1 Ulon = Ulao = 0L where g,(0) = h.,-(0) = 0, 9,, h,- are C1, strictly increasing for i = 1,2. 4. 1 Uniqueness The main theorem is the following: 43 44 Theorem 4.1 Suppose there exist k1,k2 > 0 such that a — A1 — 92(k2) > 0,d — h1(k2) < 0,d — A1— h2(k1) > 0,0. — g1(l€1)< 0 (272d 30_ 9 ,_ 4 inf(9'1) iani) > SUP-—g-‘—(sup(9£))2+sup—d h‘ (sop(hL'L))2 [“11 [0M] 9d—h2Ik11—h1 9L-gLIk21—LL + 2 81113608111363)- Then (4.1) has a unique coexistence state. PROOF. By the Maximum Principle, 6.14,, < k2. But, since go is increasing, 92(0d_h,) < g2(k2), and so A1(A + (a — go(dd_h,))1 > A1(A + (a — 92(IC2))I) > 0, since a—g2(k2) > A1 > 0. Similarly, we have A1(A+(d—h2(60_g,))1) > 0. Hence, by Theorem 1.1 in [9], (4.1) has a positive solution. We concentrate on the uniqueness part. Suppose (u,v) is a positive solution to (4.1). Then Au + u(a — g1(u)) = ug2(v) > 0 in {2. Hence, u is a sub solution to Az + z(a — 91(2)) = 0 in S2, Zlon = 0- Any sufficiently large positive constant is a super solution to A2 + z(a — 91(2)) = 0 in S2, Zlan = 0. 45 Hence, by the super-sub solution method, (42) u S 901,1. Similarly, we can get (4.3) v S gd—hl- For sufficiently small 6 > 0, €A6d_h1 + fed—m(d -' h1(€6d—h1)) : E[Agd—hr + gd—h1(d _ h1(€6d—h1))l > €[A6d_h, + 6d_h1(d — h1(6d—h1))] = 0 inf2, which says that 66,14“ is a sub solution to A2 + z(d — h1(z)) = 0 in S2, Zlag = 0. Since d — h1(k2) < 0, k2 is a super solution to A2 + z(d — h1(z)) = 0 in S2, Zlag = 0. Hence, by the super-sub solution method again, 0,14,, S k2. Monotonicity of go implies that 92(9) S 92(9d—h1) S 92002)- 46 Hence, M + u(a - 92092) - 91(0)) S Au + u(a - 91(U) - 9201)) = 0L which means that u is a super solution to A2 + z(a — 92(192) — gl(z)) = 0 in S2 Zlon = 0. Let d1 be the first eigenvector of Az+A12=0inS2, Zlag = 0. Then for sufficiently small 6 > 0, a —' 920132) — /\1— 91(6991) > 0 in Q and A6951) + 6451(0 — 92(k2) ‘- 91(€¢1)) = flA¢1 + (151(0 — 92(k2) — 916951») > €(A¢1 + A1951) = 0- 47 i.e., 6451 is a sub solution to AZ + 3(9 — 92(k‘2) — 91(2)) = 0 in 9, 2'39 2 0. Hence, by the super—sub solution method again, (4.4) 00—92(k2)—91 S u. Similarly, we can get (4'5) 6d~h2(k1)-h1 S 21' From (4.2) to (4.5), we have 60‘92(k2)—91 S u < 60-912 (4.6) 9d-h2(k1)—h1 S U S air—h,- Now, we are ready to prove the uniqueness result. Suppose (u1,v1) and (u2,v2) are positive solutions to (4.1). Let p = ul — U2 and q = v1— v2. Then Ap + p(a - g1(U1)— 92011)) = Aul — A122 + (a — 91(U1) - 92(v1))(U1- u2) = —Au2 — 122(0, — g1(u1) — 92(v1)) : _ALL, — ug(a — 91(u2) — 92012) + 91(U2) +9205) — 91(91) — 92011)) 48 = —u2(gl(u2) -— g1(u1) + 92(112)‘ 92(Ul)) = —u2(g§(5:)(-p) + 93(i)(-II)) = u2099101?) +qgé(i)) in Q, where 13,5: are from Mean Value Theorem depending on u1,u2, v1, v2. Hence, (4-7) Ar + (a - 91(U1) — 92021));9 — U2(pg’1(i‘) + (19301)) = 0 in 9. Similarly, we can get (4-8) Aq + (d - h1(v2) - h2(U2))q - v1(qh'1(fl) + 2213(9)) = 0 in 9. where y, y are from Mean Value Theorem depending on u1,v1,u2, v2. Since A1(a — gl(u1) — g2(v1)) = 0, by the variational characterization of the first eigenvalue, (4.9) [Id—A2 — (a — gl(u1) — go(v1))z) 2 0 for all z E C262). Similarly, we can get (4.10) )9 LII—Aw — (d — h1(v2)- h2(u2))w) 2 o for all w 6 02(5)). 49 From (4.7) and (4.8) we have -pAp - (a - 91(u1) - 92(v1))r2 + 11219099653) + (19505)) = 0 -qu - (d - h1(v2) - hL(U2))q2 + v1q(qh’1(:&) + ph’2(§)) = 0 in 52. Using (4.9) and (4.10) we have [QIuLpngan + (Lg-2(a) + v1q(qh’1(37) + phé(fl))l s o. i.e., [DIWP291(5?) + (“293973) + ”1h§(9))PQ + viq2hi(9)l S 0. Using the quadratic discriminant, q E 0 if (uggg(x) + vlh'2(y))2 — 4u2vlg'1(x)h'1(y) < 0 in 52, P which is equivalent to 21—2 21—2 I—I— I~I~ - “292(3) “l" ”1h2(9) + 2u29192($)h2(9) - 4u2vl91($)h1(9) < 0 1n 9- i.e., 4U2v19'1(5r)h’1(3?) > 139302? + 1294(3))? + 2U2v19§(i‘)h§(§) in 9 if and only if I ~ I ~ u I — U I — I - I — ' 491(xlh1(9) > 11392073)2 + 512512602 + 292(x)h2(y) 1n 9- 50 It is easy to see that if the inequality regarding gl,g2,h1 and ho in the theorem holds, then (4.1) is obviously true, consequently p E q E 0. The uniqueness is proved. From the above theorem, we can see that for (4.1) to have uniqueness coexis- tence state, the self limitation rates g1, hl should be quite large and the competition rates go, ho should be relatively small. 4.2 Uniqueness with small perturbation for re- production rates This section will be a generalization of section 3.2. In section 3.2, we got some uniqueness result in a neighborhood with general competition rates. In this section, we try to get a uniqueness result in a neighborhood with general self limitation and competition rates. We consider the model studied in the previous section Au + u(a - 91(U) — 92(0)) = 0 (4-11) Av + v(d - h1(v) — h2(u)) = O in S2, ulag = 0'39 2 O. The following is the main theorem. Theorem 4.2 Suppose a > A1(go(6d_h,)), d > A1(h2(90_g,)), and 91(2) > a, h1(z) Z d for z > co for some positive constant co. Suppose (4.11) has a unique 51 coexistence state (u, v) and the Frechet derivative of (4.11) at (u,v) is invertible. Then there is a neighborhood V of(a, d) in R2 such that if (ao, do) E V, then (4.11) with (a, d) = (ao, do) has a unique coexistence state. PROOF. Since the Frechet derivative of (4.11) at (u, v) is invertible, by the Im- plicit Function Theorem, there is a neighborhood V of (a, d) in R.2 and a neigh— borhood W of (u, v) in [C(‘,?‘L“‘(f—2)]2 such that for all (ao, do) E V, there is a unique positive solution (uo, vo) E W of (4.11). Suppose the conclusion of the theorem is false. Then there are sequences (a,,,d,,,u,,,v,,), (an,dn,u;,v;) in V x [Cg+a(i2)]2 such that (un, v,,) and (u,"‘,, 11;) are the positive solutions with (a, d) = (an, d") and (umvn) aé (U2, v3) and (amdn) ——> (a, d). By the standard elliptic theory and esti- mates 4.6, (umvn) —> (u,v), (ufuvg) —+ (u*,v"‘), and (u,v) (u*,v*) are solutions of (4.11). Claim 11 > 0,17 > 0,u” > 0,v* > 0. It is enough to show that ii and 27 are not identically zero because of the Maximum Principle. Suppose not, then by the Maximum Principle again, one of the following cases should hold: (1) a is identically zero and o is not identically zero. (2) u is not identically zero and v is identically zero. (3) ii is identically zero and i7 is identically zero. (1) Suppose ii is identically zero and o is not identically zero. Let a}, = —“B—— v2, = vn for all n E N. Then llunlloo , Allin + UTn(an — 91 (U11) _ 926611)) : O . in 52. A511 + v~n(dn — hl(v~n) _ h2(un)) : 0 52 From the elliptic theory, u}, -> a, and A9 + 9(0 ‘- 92(9)) = 0 . 1n (2 since g1, ho are continuous and 91(0) = h2(0) = 0. i.e., a = A1(go(i‘))) and 27 = 0d_;,,. Hence, a = A1(go(9d_h,)), which is a contradiction to our assumption. (2) Suppose ii is not identically zero and i7 is identically zero. We use a similar argument as in case (1). Let ii}, = un and v1, = “7:2: for all n E N. Then All}, + 11;,(071 — 9101711) "' 92(1),,» : 0 . 1n S2. M, + snot, — IL,(L,.) — IL,(LL~,.)) = 0 The elliptic theory concludes that 2],, —> i), and Afi+Ma—mw»=0. 1n 9 A5 + v(d — h2(i’i)) = 0 since gg,h1 are continuous and 92(0) = h1(0) = 0. Hence, it = 00-9, and d = A1(h2(i‘i)). Hence, d = A1(h2(da_g,)), which is a contradiction to our assumption. (3) Finally, suppose both a and v are identically zero. Let ii}, = '71—‘51“: and vi, = “Tull: for alln E N. Then Ada + dn(an " 91(un) _ 92(Un)) = O , in (2. AU; + 7],,(dn — h1(’Un) — h2(un)) Z 0 53 The elliptic theory concludes that if" —> it and v2, —> a, and A21 + an 2 0 . 1n 9, Av + dz) = 0 since 91,92, h1, h2 are continuous and 91(0) 2 92(0) 2 h1(0) = h2(0) = 0. Hence, a = d = A1, which is a contradiction to our assumption. Consequently, (£1517) and (if, v“) are coexistence states for (a, d). But, since the coexistence state with reSpect to (a,d) is unique, (il,v) :2 (u‘, v“) z (u, v). But, since (umvn) # (u,,,v,‘;), it contradicts the Implicit Function Theorem. Next we give one condition to guarantee the invertibility of the Frechet deriv- ative. Theorem 4.3 Suppose (u,v) is a positive solution to (4.11), and 4 [335591) [ggglmouv > [sup(gau + sup(havr. Then the Frechet derivative of (4.11) at (u, v) is invertible. PROOF. The Frechet derivative at (u, v) is —A + 91(U)+ WSW) + 92(1)) - a ugé(v) vh§(u) —A + h1(v) + vh’1(v) + h2(u) — d 54 It is enough, from the Fredholm Alternative, to show that N (A) = 0. If -Aso + (m(a) + u91W) + 92(1)) - (1)99 + 93(v)uw = 0, —Ai,b + h.’2(u)v

[(sup(g§))u + (sup(h§))v]2, then the integrand in the left hand side is positive definite in 0. Therefore (15 E a E O. Combining Theorems 4.1, 4.2 and 4.3, we have Theorem 4.4 Suppose there exist 19th > 0 such that a > A1 + 92(k2),d > A1 + h2(k1), a — 91(k1) < 0, d — h (k2) < 0, and 4inf[o,k,](gi)inf[0,k,](h’1) > [sup(gg) + sup(h’z) sup _ag—(L—fi—“S sup(g’z) sup 937:4— + sup(h’ )]. Then there is a neighborhood V of (a, d) in R2 such that if (ao,do) E V, then (4.11) with (a, d) = (a0, do) has a unique coexistence state. PROOF. IV Since 60-91 > 00—92(kzl—91 and Od-hi > gd-h2(k1)-h1' 55 From 60-9, < k1, 6,14,, < kg and the monotonicity of 92,}12, we have a > A1 + 92092) 2 A1(g2(6d—h1))2 d > A1+ h2(k1) 2 A1(h2(6a—gl))- 4 [5313591) [3gf](h1 1) 6 [sup(g2) + sup(h2 )SUP 0 d 0— ”9209) 91 ) O'- [sup(géfl2 sup + sup(g2) S1190732) 911— h2(k1)— —h1 60—91 6d—h1 I I + 8111) 5———— sup —— sup(g2) sup(h2) d h2(k1) hl 96—h, 60—92 (k2)-—91 a—g [sup(h')l2 811p [sup(géfl2 811p + 23up(92) sup(h; ) 911— h2(lc1)— —h1 94—h, [1111122112 sup 2 a-92(k2)—91 6 - HSUD(92) 811p ———6 d h2(1'91 + sup(h;2 )] Therefore, by Theorem 4.1, (4.11) has a unique coexistence state (u, v). Furthermore, by the estimate of the solution in the proof of Theorem 4.1, > Z 413‘tf1(9’1)11‘1‘tf1(h’1) 96—h 60- -92(k2)—91 [SUp(g-’2) + suI)(h’2)- :Hsup(92): + sup(héfl- [sup(gé) + sup(hé) sup llsup(g2) 811p 6d- h2(k1)— —h1 + sup(hf )] 56 Hence, 4 [grllcf1(g’1) [grifl(h’1)uv > [sup(92)u + SUp(h’2)vlz- The Frechet derivative of (4.11) at (u, v) is invertible from Theorem 4.3. Conse- quently, the theorem follows from Theorem 4.2. 4.3 Uniqueness in some domain of reproduction rates This section will be a generalization of section 3.3, where some uniqueness result was established when the reproduction rates are in some domain with general competition. In this part, we consider the similar problem for (4.11). We try to find a region of reproduction rates a and d that guarantees the existence of a unique positive solution to (4.11). In this section, we assume limHOG 91(2) :— lim,,_,c,o h1(z) 2 00. The following is the main theorem. Theorem 4.5 Let ghhl E C2,limz_m 91(2) 2 limzsoo h1(z) = 00. Let I‘ be a closed, convex region in R2 such that for all (a, d) E F, a > /\1(gg(0d_h,)) and d > )11(h2(00_g,)). Let BLI‘ = {(Ad,d) E FIFor any fixed d,)1,1 = inf{a|(a, d) E I‘}}. Suppose that (4.11) has a unique positive solution for every ((1, d) E BLI‘. Suppose that for all (a, d) E F, the Frechet derivative of (4.11) at every positive solution to (4.11) is invertible. Then for all (a, d) E I‘, (4.11) has a unique positive solution. 57 PROOF. For each fixed d, let Ad 2 sup{a : (a, d) E I‘} and Ad 2 inf{a|(a, d) E I‘}. We need to show that for every a such that Ad 3 a 3 Ad, (4.11) has a unique positive solution. Since (4.11) with (a, d) 2 (Ad, (1) has a unique positive solution (u, v) and the Frechet derivative of (4.11) at (u, v) is invertible, by theorem 4.2, there is an open neighborhood V of (Ad,d) in R2 such that if (ao,do) E V, then (4.11) with (a, d) = (ao,do) has a unique positive solution. Let A, = sup{A 2 Ad : (4.11) has a unique coexistence state for Ad 3 a S A}. We need to show that A, 2 Ad. Suppose A, < Ad. From the definition of A,, there is a sequence {An} such that A" —> A; and there is a sequence (umvn) of the unique positive solution of (4.11) with (a, d) = (A,,, d). Then by the Elliptic theory, there is (no, v0) such that (umvn) converges to (u0,vo) uniformly and (u0,v0) is the solution to (4.11) with (a, d) = (A,, d). We claim that uo is not identically zero and v0 is not identically zero. Suppose this is false. Then by the Maximum Principle, one of the following cases should hold: (1) uo is identically zero and v0 is not identically zero. (2) v.0 is not identically zero and v0 is identically zero. (3) Both uo and v0 are identically zero. The argument is similar to what we had in the previous chapter. (1) Suppose uo is identically zero and v0 is not identically zero. Let u}, 2: —“*—- and 27,, = v,, for all n E N. Then 1 Hunlloo A1171: + Un(/\n _ 91 (“11) _ 92(1L1)) = O . 1n 9. A6,, + 2:,(11 — 111(23,)— h,(12,,)) = o 58 We know u}, —> it from the elliptic theory, and All + Lt()\s — gg(’Uo)) I O . 1n 0 AUG + 00(d — h1(’Uo)) = 0 since 91,92, h1,h2 are continuous and 91(0) 2 h2(0) = 0. Hence, v0 = 6,4,, and A, = A1(gg(v0)). i.e., A, = A1(gg(64_h,)) < Ad which is impossible. (2) Suppose uo is not identically zero and v0 is identically zero. Let 2L, = un and 17,, = ”—v‘z’lfi: for all n E N. Then Au}, + 21",,(An — gl(if,,) — g2(v,,)) = 0 . 1n 9 Av}, + 17,,(d -— h1(v,,) — h2(if,,)) = 0 Again v), —+ i} by elliptic theory, and Aug + ug(A, — 91(u0))= 0 _ 1n {2 since 91,92, h1,h2 are continuous and 92(0) = h1(0) = 0. Hence, no 2 6,1,-91 and d = A1(h2(uo)). i.e., d = A1(h2(6,\,_g,)) < A1(h2(6,\a_gl)) which is impossible, since (Ad,d) E I‘. (3) Suppose uo 2 v0 = 0 59 Let if" = 44— 21116211,: —”H— for allnEN. Then llunlloo “'UHHOO Ada + “m(An _ 910122) — 92011;» = 0 All", + ’U~n(d — h1(’Un) — h2(u,,)) = 0 in Q. The elliptic theory concludes that u”, —> 11 and 22”,, —> 17, and Aa+ka=0. 1n 9 A1? + (if) = 0 since 91,92, h1,h2 are continuous and 91(0) 2 92(0) = h1(0) = h2(0) = 0. Hence, d = A, = A1, which is impossible from the hypothesis. Consequently, uo is not identically zero and v0 is not identically zero. Hence, by the Maximum Principle again, uo > 0, v0 > O in 9, that is, (uo, v0) is a coexistence of (4.11) with (a, d) = (A,, d). Since (A,, d) E I‘, by the assumption, the Frechet derivative of (4.11) at (no, vo) is invertible. Hence, by the Implicit Function Theorem, there is an open neighborhood U of A, and an open neighborhood V of (no, v0) such that if a E U, then (4.11) has a unique coexistence state in V. But, by the definition of A,, there is a sequence {A1,} Q U such that A;, —> A: and there is a sequence {(14, vj,)} of the coexistence state of (4.11) with (a, d) 2 (A;, at) such that (211,14) 1% V for all n E N. By the elliptic Theory again, u; —> 123,121, ——> v6 and from the same argument above, (11,, v6) E V is also a coexistence of (4.11) with (a, d) = (A,, d). Since (A,, d) E P, by the assumption again, the Frechet derivative of (4.11) at (213,125) is invertible. Hence, by the Implicit Function Theorem again, 60 there is an open neighborhood U’ of A, and an open neighborhood V’ of (us, v6) such that if a E U’, then (4.11) has a unique coexistence state in V’. Consequently, there are points in the left side of A, such that (4.11) has two different coexistence states. That is a contradiction to the definition of A,. Hence, A, 2 Ad and the theorem follows. By the similar argument, we can prove the following: Theorem 4.6 Let g1,h1 E Cr",limz_,00 91(2) : limzsoo h1(z) 2 00. Let I‘ be a closed, convex region in R2 such that for all (a, d) E I‘, a > A1(g2(0d_;,,)) and d > A1(h2(6a_g,)). Let 83F = {(Ad,d) E FIFor any fixed d, Ad 2 sup{a|(a,d) E F}}. Suppose that (4.11) has a unique positive solution for all (a, d) E 63F. Suppose that for all (a, d) E I‘, the Frechet derivative of (4.11) at every positive solution to (4.11) is invertible. Then for all (a, d) E F, (4.11) has a unique positive solution. PROOF. For each fixed d, let A, = inf{a : (a,d) E F} and Ad : sup{a : (a,d) E F}. By the similar argument in Theorem 4.5, we can show that for every a such that Ad 3 a 3 Ad, (4.11) has a unique positive solution, and it follows. Actually, if we consider the result in section 2, then we can extend the region F in Theorem 4.5 and Theorem 4.6 to an open set including F to guarantee the unique coexistence state. By the similar proof as in Corollary 3.21 and Corollary 3.22, we have the followings: 61 Corollary 4.7 Let gl, h E 02, lim,_,,o 91(2) 2 limb», h1(z) 2 00. Let I‘ be a closed, convex region in R2 such that for all (a, d) E P, a > A1(gg(621)) and d > A1(h2(631)). Let 01.1“ : {(A,,d) e I‘lFor any fixed xx, = inf{a|(a,d) E I‘}}. Suppose that (4.11) has a unique positive solution for all (a, d) E OLI‘. Suppose that for all (a, d) E F, the Frechet derivative of (4.11) at every positive solution to (4.11) is invertible. Then there is an open set W in R2 such that P C W and for every (a, d) E W, (4.11) has a unique coexistence state. Corollary 4.8 Let gl, h E 02,1imz_,oo 91(2) 2 limzso, h1(z) 2 00. Let I‘ be a closed, convex region in R2 such that for all (a, d) E F, a > A,(g,(61;1)) and d > A1(h2(631)). Let 0,1“ = {(Ad,d) e I‘lFor any fixed d,Ad = sup{a|(a,d) E I‘}}.Suppose that (4.11) has a unique positive solution for all (a, d) E 03F. Suppose that for all (a, d) E I‘, the Frechet derivative of (4.11) at every positive solution to (4.11) is invertible. Then there is an open set W in R2 such that I‘ C W and for every (a,d) E W, (4.11) has a unique coexistence state. CHAPTER 5 Combined self-limitation and competition This chapter will be a generalization of chapter 4. In chapter 4, we obtained some unique result in general self-limitation and competition rates. In this chapter, we try to see what happens when the self-limitation and competition are combined. We consider the model f Au+ u(a — g(u,v)) = 0 in (2 (5.1) 1 A2 + v(d — h(u, 2)) = o ’ 1 “lat: = ’Ulao = 01 where g(O, 0) = h(O, O) = 0, g, h are Holder continuous up to the second order in every compact set in R2, all the partial derivatives of g and h are positive. 62 63 5. 1 Uniqueness The main theorem is the following: Theorem 5.1 Suppose there exist k1,k2 > 0 such that a — A1 — g(O, k2) > 0,d — h(0,k2) < 0,d— A1 — h(k1,0) > 0,d — g(k1,0) < 0,61,71,61 . 39 . 5h (922—950) 59 2 6d—h(0,) 5h 2 4lgf(6u)lgf(8v) > SUP 6d—h(k1,)(sup(6v)) +5111) 0a_g(,k,)(sup(6u)) 89 6h + 2sup<55>sup(,—u>, where B = [0,191] x [0, k2]. Then (5.1) has a unique coexistence state. PROOF. Existence part was already established by Theorem 3.3 in [10]. We concentrate on the uniqueness part. Suppose (u, v) is a positive solution to (5.1). By the Mean Value Theorem, there is v* such that g(u,v) = g(u,0) + Wu Then Au + u(a — g(u, 0)) = Wuv > 0 in 9. Hence, it is a sub solution to Az + z(a — g(z,0)) : 0 in Q, 2'39 2 0. Any sufficiently large positive constant is a super solution to A2 + z(a — g(z,0)) = 0 in Q, 2'5”) 2 O. 64 Hence, by the super-sub solution method, (5.2) u S 6a_g(,0). Similarly, we can get (5.3) ’U S 6d—h(0,)- For sufficiently small 6 > O, 6A6d_h(o,) + €6d—h(0,)(d — h(02 €6d—h(0,))) = €[A6d-h(0,) + 6d-h(0,)(d — h(O: 60d—h(0,)))] > ClAgd—MQ) + 6d-h(0,)(d _ h(O, 6d—h(0,)))] : 0 inQ, and so 66,1440) is a sub solution to Az + z(d — h(O, 2)) = 0 in Q, 2'39 2 0. Since d — h(O, 10;) < 0, k2 is a super solution to Az + z(d — h(O, 2)) = O in Q, 2'39 2 0. 65 Hence, by the super—sub solution method again, 6,440,) S k2. Hence, g(u, ’U) S g(u, 6d—h(0,)) S g(u, k2) since g(u, z) is increasing. Hence, Au + u(a — g(u, 192)) 3 Au + u(a — g(u, v)) =0inQ. Hence, u is a super solution to A2 + 2(a — g(z, k2)) = 0 in Q, Zlag 1‘ 0. Let d1 be the first eigenvector of Az+A1z:O inQ, Z'ag = 0. Then for sufficiently small 6 > 0, a—g(6<,b1,k2)—- /\1> 0 in Q, and A(€451) + 6(151(0 — 9(€¢12k2)) = €[A<151+ ¢1(a — 9(€¢11k2))l 66 > 6(Aq51 +A1¢1) =0 in 9. Consequently, eqbl is a sub solution to A2 + z(a — g(z, 152)) = O in Q, 2'39 2 0. Hence, by the super-sub solution method again, (54) 6a-g(,k2) S U“ Similarly, we can get (5-5) 6d—h(k1,) S ”U. From (5.2) to (5.5), we have 6a— ,k S U S 6a— , 1 (5.6) 9( 2) 9( 0) (9 91141021,) S U S Now, we are ready to prove the uniqueness result. Suppose (u1,v1) and (u2,v2) are positive solutions to (5.1). Let p : ul — U2 and q 2 ’U1 — U2. Then AP +140 — g(ul, 711)) = AUl " AM + (a — g(u11v1))(u1— U2) 2 —Au2 — U2(Cl —— g(u1,v1)) = -Au2 — ug(a — g(u21 ’02) + 904217)?) 67 —g(u1,v1)) : —u2(g(u2, U2) — g(u11v1)) : —u2(g(u2,v2) — 90111712) + g(u11v2) "- g(u1,v1)) Bg(x,v2) )+ 59(U11i)(_q)) : —u2( 0v (—p 8v ('3 ~, 0 , " , : rug,(p—g—((,;l:u——P—2-2 + gig—1:32) 1n (2, where fit, 3‘: are from Mean Value Theorem depending on ul, “12,01,712. Hence, 89(53, v2) + Bg(ul,x) T q 22 PM” (5.7) Ap + (a —— g(u1,v1))p — ug(p Similarly, we can get 6h(ga ”1) + q3h(u2, 37) (58) Ag -+- (d — h(Ug, 112))(1 — v1 (p 0U av )=0 inQ, where g, 37 are from Mean Value Theorem depending on u1,v1,u2,v2. Since A1(a — g(u1,v1)) = 0, by the variational characterization of the first eigen- value, (5.9) /02(—Az — (a — g(u1,v1))z) Z 0 for all z E 02(0). Similarly, we can get (5.10) /Qw(—Aw — (d — h(u2,v2))w) 2 o for all 12 6 02(5)). 68 From (5.7) and (5.8) we have —pAp—:0 11.12 -qu-(d h(”Lt2,v2))q +v1q(p MHz—WM): 0 Using (5.9) and (5.10) we have 5’43}, v1) + BMW, 3]) ag< x ,UQ) 69(1‘111‘) [he p(—-——p +q——) +v1q(p 621 (1 00 au (9v Hence, 0 ~ 8 ‘ 6h ”, 0h , ‘ [m g(zz:,'v2)p2+(u2 9611,22) +01 (6 120m,”1 (U2 y)q2 a 8n 8v 0n Therefore, p E- q E 0 if we can show that 696(11):?) +1} 3h(y,v1)2 39(53,v2) ah(u2ag) (9v v1 flu )2— mm Bu 3v <01nQ, (“2 which is true if ug(aqtul ,x))2 + ’U 2( 6h(&v1))2 + 211.2111 8g(g1:,i) ahgfl) —4u2v1 a—ggffl 13/191352) < O in Q. i.e., 4U2U1 69(23, v) 6h(6vU2Q) > u 2( 0,9(u1 a1),a':))2+ ”2(6hg/uv12y +2u2v1 655513 ahgfim in St, 69 OI' 459(i,'v2)3h(ue.3}) > m(agéulizy + u(ah(37,012)2 (911. 6v v1 01) U2 Bu ”69%;” “$351) in n. This is the case from the hypothesis in the theorem and (5.6), the uniqueness is proved. From the above theorem, we can see that for (5.1) to have uniqueness coex- istence state, it is better that the self limitation rates are quite large and the competition rates are quite small. 5.2 Uniqueness with small perturbation of re- production rates This section will be a generalization of section 4.2, where we proved some unique- ness result in a neighborhood of reproduction rates with general self limitation and competition rates. In this part, we extend these results to combined self limitation and competition. The following is the main theorem. Theorem 5.2 Suppose a > A1(g(0,6d_h(0,))),d > A1(h(90_g(,0),0)), and g(z,0) Z a,h(0,z) Z d for z 2 Co for some constant co > 0. Suppose (5.1) has a unique coexistence state (u,v) and the Frechet derivative of (5.1) at (u,v) is invertible. Then there is a neighborhood V of (a, d) in R2 such that if (ao, do) 6 V, then (5.1) with (a, d) = (do, do) has a unique coexistence state. 70 PROOF. Since the Frechet derivative of (5.1) at (u, v) is invertible, by the Implicit Function Theorem, there is a neighborhood V of (a, d) in R2 and a neighborhood W of (u, v) in [CngaU—D]2 such that for all (ao, do) 6 V, there is a unique positive solution (u0,v0) E W of (5.1). Suppose the conclusion of the theorem is false. Then there are sequences (on, dn, un, v,,), (a,,, ct", u:,, v;) in V x [C(,2+‘"(S—2)]2 such that (un, vn) and (u‘ 2);) are the positive solutions with (a, d) = (on, d") and (umvn) ¢ 11’ (u* v*) and (amdn) —> (a,d). By the standard elliptic theory, (umvn) —> (21,27) and (u;,v,‘;) —-> (u*,v*) in 02’“, and (22,27) (u‘,v") are solutions of (5.1). Claim 11 > O, 27 > O, u* > 0, v* > 0. It is enough to Show that ii and v are not identically zero because of the Maximum Principle. Suppose not, then by the Maximum Principle again, one of the following cases should hold: (1) a is identically zero and o > 0. (2) a > O and o is identically zero. (3) a is identically zero and v is identically zero. (1) Suppose a is identically zero and 27 > 0. Let 11,, = W, v}, 2 v7, for all n E N. Then 71 00 Au~n + 11:40” — g(uni 611)) 2 0 . in 9 Av}, + v”,,(dn — h(un, 22”,,» = 0 From the elliptic theory, 1L, —> a and AM] + u(a — g(0,v)) = O . in Q Av + v(d — h(O, 27)) = 0 since g,h are continuous. i.e., a = A1(g(0,v)) and ”t7 = 9d_h(o,)- Hence, a = 71 A1(g(0, 921—240,)», which is a contradiction to our assumption. (2) Suppose a > 0 and 27 is identically zero. We use a similar argument as in case (1) and details are omitted. (3) Finally, suppose both a and v are identically zero. Let 22;, = —‘—‘n— and 23;, = —"n— for allnEN. Then llurtlloo “vnlloo Au}, + if,,(a,, —— g(un, vn)) = O in 9. Av}, + '17,,(dn — h(un, vn)) = 0 The elliptic theory concludes that u}, —> a and 27,, —> i), and A1] + a1] = 0 in 0 A1? + (123 = 0 since g, h are continuous and g(0,0) = h(O, O) = 0. Hence, a = d = A1, which is a contradiction to our assumption. Consequently, (21,27) and (u*, v“) are coexistence states for (a, d). But, since the coexistence state with respect to (a, d) is unique, (21,27) 2: (u*,v*) : (u, v). But, since (um vn) # (u;,v,‘;), it contradicts the Implicit Function Theorem. Now we turn out attention to get conditions to guarantee the invertibility of the Frechet derivative. 72 Theorem 5.3 Consider the model Au + u(a — g(u, v)) = O in Q. { Av + v(d — h(u,v)) = O \ UlaQ = ’Ulag = 0. Suppose (u, v) is a positive solution to the above model. If 4inf3 flgi—fl infB 6%?qu > [sup 9%:My2u + sup QEéZMylvP, where B = [0,k1] x [0, k2], then the Frechet derivative of the above model at (u, v) is invertible. PROOF. The Frechet derivative at (u, v) is vw’aziz —A+h(u,v)+vaié(;vM—d We need to show that N(A) = {0}. If -Aso + (g(u, 11) + u—l—lagaf,” — (1)90 + 44396:” ml = 0, —A1,b + va + (h(u, v) + v—J—Mhazw — d)i/2 = 0, then fallvwl2 + (g(u, v) + 118—95324 - CW2 + WWI/J] = 0, faHWIZ’ + Q’aéiflvcpw + (h(u, v) + 716—3934") — (1)2122] = 0. Since A1(g(u,v) — a) = /\1(h(u, v) — d) = 0, falleOI2 + (g(u,v) — a>902] 2 0, anViDl2 + (h(u, 1)) - 601/22] 2 0. 73 Hence, Jaw—4%” 902 + My 119011)) S 0, fd—Lflhat" w + My? 2w?) s 0. Hence, ffllua—ggfvlgfl + (fig—flu + flngw + vaq S 0. Hence, if 4inf3(a—gg%l)inf3(flg:—’fl)uv > [(sup(@§3%l»u + (sup(fléa—Efl))v]2, then the integrand in the left side is positive definite form in 52, which means go E w E 0. Therefore, the above Frechet derivative A is invertible. Now, we try to get another uniqueness result. We continue to consider the same model r Au + u(a — g(u,v)) = 0 in 9 (5-11) i Av + v(d — h(u,v)) = 0 \ Ulan = vlao = 0- Theorem 5.4 Suppose there emist k1,k2 > 0 such that a > A1 + g(0,k2), d > A1+ h(k1,0),a — g(k1,0) < 0,d — h(0,k2) < 0, and 4infB giggzlinfg @3322 > [sup @g—‘yl + sup 9%?!) sup 6931101pr 6—9552?) sup m + sup 956333], where B = 0-9(»k2) 9d—h(k1.) [0,191] x [0, k2]. Then there is a neighborhood V of (a, d) in R2 such that if (a0, do) 6 V, then (5.11) with (a,d) = (ao,do) has a unique coexistence state. PROOF. From 0a_g(,0) < k1, 60,440,) < kg, and the monotonicity of g(O, ),h(,0) we have a > A1 + g(O, 182) Z )‘l(9(026d—h(0,)))i d > A1 + h(k1,0) Z A1(h(60._g(‘0),0)). 4 igf fight, 31) inf 0km 31) 63: B By 74 09(x,y) 0h(x,y) 9d_,,(0) 905,3!) 190-9(0) > su ——+su ———su ——’—su su ’ l p 63, am 6,_g(,,,,,” p by 0.14.0.1, 6h , + sup _5’9: 9)] 6(1— , h 1 , : [sup 59(23,y)]28up gm) + sup 09(23 y) sup 0 (2 y) ' 6d—h(k1,) By 0113 00—4 0) 6d—h(0) 69(1‘9 y) ah($a y) +811 -—J—,— ———"—Sll —,——SU ——‘,.—-— p 6‘1"“li ga—gbkz) p dy p 0:12 +[su 0h($’y)]zsu ———6d—h(0’) ax 6a—g(,k2) 60- , h , 2 [sup 09(x,y)]2sup gm) + 2811p 09(23 y) sup 0 (It y) 03; 6d—h(k1,) all abbot!) 2 Oar-hm.) +[sup 8:1: ] su 9a—g(,k2) Since 9a—g(.0) > (la—gm» 9d—h(0,) > ltd—hon.)- 0h B 0:12 B 0y 09(r, y) 6h(:c, y) 944(0) 59(3) y) a—g( 0) > su ———+su ——,——su ——’— su su ' [ p 8y p ()2: p 6a—g(,k2)][ p 0 p gal—hum) h . +sup _3 E9: 31)] 60— v a 1 ah I ’ : sup gm) (sup 9(1 v))2 +sup 9(13 y) sup (2: y) d—h(k1,) 8y By 02: 60— (O) 6d—h(0) 69(3)?!) 6M3)?» +su —3’——su ——-—’—s —— —— p gd-Mkn) p 9a—gtk2) p a?! p 590 +8111) 6d—h(0,) (SH 611(2)) y) )2 60-9(.k2) 6:13 0.1. 0) 89(36 y) 09(93 y) 0W? U) > su —g(’ su ——’-— 2+2su ——'——1——su ————1'—‘— _ p 9d—h(k1,)( I) 6y ) I) 6y I) 02 +sup 9d—h(o,) (su (9h(:c,y))2 a-9(.k2) 593 ' Therefore, (5.11) has a unique coexistence state (u, v) from Theorem 5.1. Further- 75 more, by the estimate of the solution in the proof of Theorem 5.1, 09(26, y) inf (War, 31) 4 igf 827 B 03/ > [sup Qgiajli) + sup gig—3L) sup bid—:g’fiflsup 6—922—3’2 su gig-fit); + sup -ah—(;;—y)] Z [sup 9%?9 + sup @é—Z-ylgflsup 29%?3 + sup W]. Hence, 4 igf @721) igf wuv > [sup 69233’ mu + sup Bhg: y)v]2. Hence, the Frechet derivative of (5.11) at (u,v) is invertible from Theorem 5.3. Hence, the theorem follows from Theorem 5.2. 5.3 Uniqueness in a region of reproduction rates This section will be a generalization of section 4.3, where we got some uniqueness result in some domain with general self-limitation and competition rates. In this section, we extend these results when the self-limitation and competition are com- bined. 76 Consider the model r Au+u(a—g(u,v)) = O in Q (5.12) i Av + v(d — h(u, v)) = O \ ”(1'39 2 ’UlaQ = 0. We try to find a region of reproduction rates a and d that guarantees the existence of unique positive solution to (5.12). The following is the main theorem. Theorem 5.5 Let g(z,0), h(0,z) E C2,limz_+00 g(z,0) ——— limz_,oo h(O, z) = 00. Let I‘ be a closed, convex region in R2 such that for all (a, d) E I‘, a > A1(g(0, 9d—h(o,))) and d > A1(h(00_g(,0),0)). Let BLI‘ ——— {(Ad,d) E FlFor any fixed d,Ad = inf{a|(a,d) E F}}. Suppose that (5.12) has a unique positive solution for every (a,d) E (9LT. Suppose that for all (a,d) E I‘, the Frechet derivative of (5.12) at every positive solution to (5.12) is invertible. Then for all (a, d) E I‘, (5.12) has a unique positive solution. PROOF. For each fixed d, let Ad 2 sup{a : (a, d) E F} and Ad 2 inf{a|(a, d) E F}. We need to show that for every a such that Ad 3 a S Ad, (5.12) has a unique positive solution. Since (5.12) with (a,d) 2 (Ad, d) has a unique positive solution (u, v) and the Frechet derivative of (5.12) at (u, v) is invertible, by theorem 5.2, there is an open neighborhood V of (Ad,d) in R2 such that if (ao,do) E V, then (5.12) with (a,d) = (ao,do) has a unique positive solution. Let A, : sup{A 2 Ad : (5.12) has a unique coexistence state for Ad 3 a S A}. We need to show that A, 2 Ad. Suppose A, < Ad. From the definition of As, there is a sequence (A,,} such 77 that A,, —> A; and there is a sequence (umvn) of the unique positive solution of (5.12) with (a, d) = (A,,, d). Then by the Elliptic theory, there is (uo, vo) such that (u,,,v,,) converges to (u0,vo) uniformly and (uo, v0) is the solution to (5.12) with (a, d) = (A,, d). We claim that uo is not identically zero and v0 is not identically zero. Suppose this is false. Then by the Maximum Principle, one of the following cases should hold: (1) uo is identically zero and v0 is not identically zero. (2) no is not identically zero and v0 is identically zero. (3) Both uo and v0 are identically zero. The argument is similar to what we had in the previous chapter. (1) Suppose uo is identically zero and v0 is not identically zero. Let 2L, 2 f“— and 1],, 2 v7, for all n E N. Then ll nlloo Au~n + UTn(An _ “7112,1522”: O in 9. A67: + 1711((1 — h(uni 1):”) Z 0 We know u} —> a from the elliptic theory, and Aii + u(A, —- g(0,v0)) = 0 . in Q Avg + v0(d — h(0,v0)) = 0 since g,h are continuous. Hence, v0 = 9.1440,) and A, = A1(g(0,v0)). i.e., A, = A1(g(0, 6d_h(0,))) < Ad which is impossible. (2) Suppose uo is not identically zero and v0 is identically zero. 78 Let 21,, = u" and 2],, = ——"D— for all n E N. Then ll‘vnlloo M, + lino, — g(u,, v,» = o in (2. Av} + v~n(d — h(u'n, vn)) = 0 Again v2, —> 27 by the elliptic theory, and Alto + U0()\3 — g(uo, 0)) = O . 1n 9 since g,h are continuous. Hence, uo = $3-9m) and d : A1(h(u0,0)). i.e., d = A1(h(6,\,_g(,o),0)) < A1(h(6Ad_g(’0),0)) which is impossible, since (Ad,d) E I‘. (3) Suppose uo 2 v0 = 0. Let 22;, = W and on: m: for allnE N. Then Allin + dn()‘n — g(un2vn)) : O in 9. Av}, + v",,(d — h(un, vn)) = O The elliptic theory concludes that 25,, -> it and 17,, —-> v, and Au+ka=0 inQ Av+dv=0 since 9, h are continuous and g(O, 0) = h(0,0) = 0. Hence, (1 = A, = A1, which is impossible from the hypothesis. Consequently, uo is not identically zero and v0 is not identically zero. 79 Hence, by the Maximum Principle again, uo > 0, v0 > O in 9, that is, (an, v0) is a coexistence of (5.12) with (a,d) = (A,,d). Since (A,, d) E I‘, by the assumption, the Frechet derivative of (5.12) with (a, d) = (A,, d) at (uo, v0) is invertible. Hence, by the Implicit Function Theorem, there is an Open neighborhood U of A, and an open neighborhood V of (uo,v0) such that if a E U, then (5.12) has a unique coexistence state in V. But, by the definition of A,, there is a sequence {AL} Q U such that A; —+ A: and there is a sequence {(u’ v’ )} of the coexistence state of (5.12) with (a, d) : (A1,,d) such that (24,24) E V for all n E N. By the Elliptic Theory again, ufi, —> ug, vi, —+ v6 and from the same argument above, (1L6, v6) E V is also a coexistence of (5.12) with (a, d) = (A,,d). Since (A,,d) E I‘, by the assumption again, the Frechet derivative of (5.12) at (us, v6) is invertible. Hence, by the Implicit Function Theorem again, there is an open neighborhood U’ of A, and an open neighborhood V’ of (u6,v(’,) such that if a E U’, then (5.12) has a unique coexistence state in V’. Consequently, there are points in the left side of A, such that (5.12) has two different coexistence states. That is a contradiction to the definition of A,. Hence, A, _>_ A" and we are done. By the similar argument, we can prove the following: Theorem 5.6 Let g(z,0),h(0, z) E 02,1im,_,oo g(z,0) = limznoo h(O, z) = 00. Let F be a closed, convex region in R2 such that for all (a,d) E F, a > A1(g(o,s,_,,(,,)) and d > A1(h(0,_,(,,,0)). Let em = {(A,,d) e FlFor any fixed d,A" 2: sup{a|(a,d) E I‘}}. Suppose that (5.12) has a unique positive solution for all (a,d) E 83F. Suppose that for all (a,d) E I‘, the Frechet derivative of (5.12) at every positive solution to (5.12) is invertible. Then for all 80 (a, d) E F, (5.12) has a unique positive solution. PROOF. For each fixed d, let Ad 2 inf{a : (a, d) E I‘} and Ad 2 sup{a : (a,d) E I‘}. By the similar argument in Theorem 5.5, we can show that for every a such that Ad 3 a 5 Ad, (5.12) has a unique positive solution, and the theorem follows. Actually, we can extend the region I‘ in Theorem 5.5 and Theorem 5.6 to an open set including F to guarantee the unique coexistence state. By the similar proof as in Corollary 3.21 and Corollary 3.22, we have the followings: Corollary 5.7 Let g(z,0), h(O,z) E 02,1imz_,oo g(z, O) = lim,_>00 h(0,z) = 00. Let 1" be a closed, convex region in R2 such that for all (a,d) E F, a > A1(g(0,9d_h(0,))) and d > A1(h(00_g(,0),0)). Let 61,1“ 2 {(Ad,d) E F|For any fixed d, Ad 2 inf{a|(a,d) E F}}. Suppose that (5.12) has a unique positive solution for all (a,d) E 611‘. Suppose that for all (a,d) E F, the Frechet derivative of (5.12) at every positive solution to (5.12) is invertible. Then there is an open set W in R2 such that F C W and for every (a,d) E W, {5.12) has a unique coexistence state. Corollary 5.8 Let g(z,0), h(O, z) E C2,lim,_,oo g(z,0) ——— limznoo h(0,z) = 00. Let I‘ be a closed, convex region in R2 such that for all (a,d) E I‘, a > A1(g(0,6d_h(0,))) and d > A1(h(0,_g(,0),0)). Let 031‘ = {(Ad,d) E FIFor any fixed d,A" = sup{a|(a,d) E I‘}}. Suppose that (5.12) has a unique positive solution for all (a, d) E 331‘. Suppose that for all (a,d) E I‘, the Frechet 81 derivative of (5.12) at every positive solution to (5.12) is invertible. Then there is an open set W in R2 such that F C W and for every (a,d) E W, {5.12) has a unique coexistence state. Apparently, Corollary 5.7 and Corollary 5.8 are generalizations of theorem 5.2. BIBLIOGRAPHY BIBLIOGRAPHY [1] R. S. Cantrell and C. Cosner, On the steady - state problem for the Volterra - Lotka competition model with diffusion, Houston Journal of mathematics, 13(1987), 337-352. [2] R. S. Cantrell and C. Cosner, On the uniqueness and stability of positive solutions in the Volterra-Lotka competition model with diffusion, Houston J. Math. 15(1989) 341-361. [3] C. Cosner and A. C. Lazer, Stable coexistence states in the Volterra-Lotka competition model with diffusion, Siam J. Appl. Math., 44(1984), 1112-1132. [4] D. 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Weinberger, Maximum principles in differential equa- tions, Prentice Hall, Englewood Cliffs, N. J ., 1967. [12] I. Stakgold and L. E. Payne, Nonlinear problems in nuclear reactor analysis, in nonlinear problems in the physical sciences and biology, Lecture notes in Mathematics 322, Springer, Berlin, 1973, 298-307. IIIIIIIIIIIIIIIIIIIIIIIIIIIIII [[1]/ll][Ill[ll][[1]]llllllllll[[[IHI