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able 4.3. (cont’d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able 4.3. (cont’d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igure 4.4. (cont’d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a15)P18()NAAJ+)!C9Y'#&9"5:bC';9>#&9"5:)457'&167495$)Y41)%;157) :'3'5$'$+)T&'5:$)45)H69;9#F)15:)HY9;"7495()NA()B-I302,72! ;0:?%?'3!%,!-%?2:;2!;22-3%,8!1',@*!A7#3#8B(!65(!C4DECCF*! ! 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A multi-species Lokta Volttera model with variable BNDD   xi (k            + 1) =xi (k)exp [bi (k) − BN DD(xi (k))xi (k) − c(k)y(k)] , n y(k + 1) =y(k)exp i=1 i = 1, 2, · · · , n di (k)xi (k) − e(k)y(k) − r(k) , BN DD(xi (k)) =xi (k)2 − xi (k)α(k), (1.3) i = 1, 2, · · · , n The system is permanent Lemma 1 If it exists l0 ∈ N such as xi (l0 ) ≤ xi (l0 + 1), then: ∀k ≥ l0 , xi (k) ≤ pi , i = 1, 2, · · · , n with pi = Ri exp [bu + (4/27)αu,3 (k)], where Ri is a root of a polynome depending on αu i and bu . i Proof: •First prove that: ∃j0 ∈ N xi (j0 ) ≤ xi (j0 + 1) =⇒ xi (j0 + 1) ≤ pi , i = 1, 2, · · · , n From (1.3) we have: 0 ≤ bi (j0 ) − BN DD(xi (j0 ))xi (j0 ) − c(j0 )y(j0 ) 0 ≤ bi (j0 ) − xi (j0 )3 + xi (j0 )2 α(j0 ) − c(j0 )y(j0 ) 0 ≤ bi (j0 ) − xi (j0 )3 + xi (j0 )2 α(j0 ) 0 ≤ bu − xi (j0 )3 + xi (j0 )2 αu i Let’s call Qi the polynome Qi (x) = bu − x3 + x2 αu i Then deltai = −4αu,3 bu − 27(bu )2 ≤ 0 i i So it exists one unique real solution Ri such as Qi (Ri ) = 0 2α It can be shown that Ri ≥ u,3 . Ri depends only on αu and bu . i Further analyses of Qi show that 0 ≤ Qi (x) => x ≤ Ri 143 Moreover Qi is maximal for x = So 2αu . 3 ñ xi (j0 ) ≤ Ri ô î ó def 2αu xi (j0 + 1) ≤ Ri exp Qi ( ) = Ri exp bu + (4/27)αu,3 = pi i 3 •Now we will prove the lemma by contradiction. So let’s assume that it exists l0 ∈ N such as xi (l0 ) ≤ xi (l0 + 1), and that ∃w0 ∈ N, w0 l0 /xi (w0 ) Let’s call w0 the smallest integer such as xi (w0 ) ˜ ˜ pi pi . Then w 0 ≥ l0 + 2 ˜ xi (w0 − 1) ≤ pi < xi (w0 ) ˜ ˜ Thus xi (w0 − 1) ≤ xi (w0 ) ˜ ˜ xi ( w 0 ) ≤ pi . ˜ We deduce from the first part of the proof: By contradiction ∀k ≥ l0 , xi (k) ≤ pi , i = 1, 2, · · · , n Theorem 1 For every solution (x1 (k), · · · , xn (k), y(k)) of system (1.3), it exists k1 ∈ N such that: ∀k ≥ k1 xi (k) ≤ pi , i = 1, 2, · · · , n (1) ∀k ≥ k1 y(k) ≤ q (2) where pi are constant depending on αu and bu , and q is a constant depending on αu , bu , rl , el , du . i i i Proof: •The previous lemma proves the first result when it exists l0 ∈ N such as xi (l0 ) ≤ xi (l0 + 1). In this case k1 = l0 . Now we assume that such an integer l0 doesn’t exist. It means that xi (k + 1) < xi (k) for all k ∈ N. As (xi (k))k is bounded under by 0, it admits a limit. Let’s call that limit xi . ¯ By taking the limit in the ith equation of system (1.3), we get lim k→+∞ Moreover Ç å bi (k) − BN DD(xi (k))xi (k) − c(k)y(k)) = 0 bi (k) − N DD(xi (k))xi (k) − c(k)y(k) ≤ Qi (xi (k)) 144 So 0 ≤ Qi (¯i ) x So we conclude that xi ≤ Ri . Note that Ri < pi . This leads to the existence of k1 ∈ N such ¯ that ∀k ≥ k1 xi (k) ≤ pi . This complete the proof of (1). •According to (1) there exists an integer k1 such that: xi (k) ≤ pi , ∀k ≥ k1 . By using the system we get for k ≥ k1 n y(k + 1) ≤ y(k) exp exp d u pi i i=1 l l − r − e y(k) = Moreover î ñ ó n i=1 d u pi i −r el l ô î el y(k) exp −el y(k) el y(k) exp −el y(k) ≤ 1 ó So we conclude that (y(k))k is bounded above, and we have: ∃k1 ∈ N ∀k ≥ k1 y(k) ≤ n i=1 du (pi ) − rl def i = q el Lemma 2 We assume that bl − c u q > 0 i (H1) If it exists an integer l0 ≥ k1 such as xi (l0 ) ≥ xi (l0 + 1), then: def ∀k ≥ l0 î xi (k) ≥ si , ˜ ó i = 1, 2, · · · , n with si = ri exp bl − p3 + p2 αl − cu q . Where ri is a root of a polynome depending on αl , ˜ i i i l u bi , c , q. Proof: •First prove that there exists constants si such that: ˜ ∃j0 ≥ k1 xi (j0 ) ≥ xi (j0 + 1) =⇒ xi (j0 + 1) ≥ si , ˜ i = 1, 2, · · · , n We assume it exists j0 ≥ k1 such as xi (j0 + 1) ≤ xi (j0 ) then from (1.3) we have bi (j0 ) − BN DD(xi (j0 ))xi (j0 ) − c(j0 )y(j0 ) ≤ 0 And from Theorem 1: Let’s call Mi the polynome bl − xi (j0 )3 + xi (j0 )2 αl − cu q ≤ 0 i Mi (x) = bl − x3 + x2 αl − cu q i 145 Then deltai = −4αl,3 bl − 27(bl − cu q)2 i i Thanks to (H1) we have: deltai < 0. So it exists one unique real solution ri such as Mi (ri ) = 0 l ri depends only on αl , cu , q. It can also be shown that ri ≥ 2α . Further analyses of Mi show 3 that Mi (x) ≤ 0 => x ≥ ri , and that Mi is decreasing for x ≥ ri . So xi (j0 ) ≥ ri . Moreover we know that xi (j0 ) ≤ pi . If pi < ri we conclude that j0 doesn’t exist. Otherwise we conclude that def xi (j0 + 1) ≥ ri exp [Mi (pi )] = si , ˜ i = 1, 2, · · · , n •Now we will prove the lemma by contradiction. So let’s assume that it exists l0 ∈ N such as xi (l0 ) ≥ xi (l0 + 1), and that it exists w such that xi (w) si . ˜ Let’s call w0 the smallest w such as xi (w0 ) si . ˜ Then w 0 ≥ l0 + 2 l0 + 1 xi (w0 ) ≤ si ≤ xi (w0 − 1) ˜ Thus xi (w0 ) ≤ xi (w0 − 1) xi (w0 ) ≥ si , ˜ We deduce from the first part of the proof: The lemma is proved by contradiction. i = 1, 2, · · · , n. Theorem 2 We assume that bl − c u q > 0 i (H1) For every solution (x1 (k), · · · , xn (k), y(k)) of system (1.3), it exists k2 ∈ N such that: ∀k ≥ k2 xi (k) ≥ pi , ˜ ∀k ≥ k2 i = 1, 2, · · · , n y(k) ≥ q ˜ (3) (4) where pi and q are constant depending on bl , αl , cu , αu , bu , el , dl . ˜ ˜ i i i Proof: •The previous lemma proves the first result when it exists l0 ∈ N such as xi (l0 ) ≥ xi (l0 + 1). Now we assume that xi (k + 1) > xi (k) for all k ∈ N. As (xk )k is bounded above, it admits a limit. Let’s call that limit xi . 146 By taking the limit in the ith equation of system (1.3), we get lim k→+∞ Ç å bi (k) − BN DD(xi (k))xi (k) − c(k)y(k) = 0 Moreover for large enough k ∈ N bi (k) − N DD(xi (k))xi (k) − c(k)y(k) ≥ bl − xi (k)3 + xi (k)2 αl − cu q = Mi (xi (k)) i Leading to Mi (xi ) ≤ 0 So, thanks to the lemma, we have: xi ≥ ri . As ri ≤ pi we know that Mi (pi ) ≤ 0. So ri ≥ ri exp [Mi (pi )] = si . So there exists k2 ∈ N ˜ def 1 such that ∀k ≥ k2 xi (k) ≥ 2 si = pi . This complete the proof of (3). ˜ ˜ •In a first time let’s assume there exists l0 ∈ N such that y(l0 ) ≥ y(l0 + 1). Then: n y(l0 ) ≥ y(l0 + 1) ≥ y(l0 ) exp So, n i=1 i=1 dl (˜i ) − ru − eu y(l0 ) i p dl (˜i ) − ru − eu y(l0 ) ≤ 0, leading to: i p n i=1 y(l0 ) ≥ dl (˜i ) − ru def i p = q ˜ eu If such an integer l0 doesn’t exist, this means that (y(k))k is increasing. As it is bounded above, it admits a limit denoted by y . By taking the limit in the (n+i)th equation of system ¯ (1.3), we get: Ç å n lim k→+∞ i=1 di (k)xi (k) − e(k)y(k) − r(k) = 0 In particular there exists k2 ∈ N such that n ∀k ≥ k2 So n ∀k ≥ k2 i=1 i=1 di (k)xi (k) − e(k)y(k) − r(k) ≤ 1 dl (˜i ) − ru ≤ i p n i=1 di (k)xi (k) − r(k) ≤ e(k)y(k) ≤ eu y(k) ∀k ≥ k2 q ≤ y(k) ˜ The proof of (4) is achieved. 147 Theorem 3 Stable positive equilibrium Let’s two positive solutions of the system be (x1 , · · · , xn , y) Û Û Û ( xi , · · · , xn , y ) and such that max 1 + 3p3 i Ç pi pi ˜ bl − c u q > 0 i å3 Ç å + 2αu p2 i n q ˜ 1−eq ˜ q + Ä ä l Then we claim that d u pi i i=1 Û lim y(k) − y (k) = 0 k→+∞ Ä ä ı lim xi (k) − xi (k) = 0, k→+∞ Proof: Let Then we have = pi pi ˜ Ç å2 pi pi ˜ , cu q å q <1 q ˜ <1 (H1) (H2) (H3) i = 1, 2, · · · , n Û xi (k) = xi (k) exp(ui (k)) Û y(k) = y (k) exp(v(k)) Û xi (k + 1) = xi (k + 1) exp(ui (k + 1)) exp(ui (k + 1)) = From system (1.3) Ç xi (k + 1) Û xi (k + 1) xi (k) exp(bi (k) − xi (k)3 + xi (k)2 α − c(k)y(k)) exp(ui (k + 1)) = Û Û Û Û xi (k) exp(bi (k) − xi (k)3 + xi (k)2 α − c(k)y (k)) Û Û Û Û xi (k) exp(ui (k)) exp(bi (k) − xi (k)3 exp(3ui (k)) + αxi (k)2 exp(2ui (k)) − c(k)y (k) exp(v(k))) Û Û Û Û xi (k) exp(bi (k) − xi (k)3 + xi (k)2 α − c(k)y (k)) Û Û Û = exp(ui (k) − xi (k)3 (exp(3ui (k)) − 1) + αxi (k)2 (exp(2ui (k)) − 1) − c(k)y (k)(exp(v(k)) − 1)) So Û Û Û ui (k+1) = ui (k)− xi (k)3 (exp(3ui (k))−1)+αxi (k)2 (exp(2ui (k))−1)−c(k)y (k)(exp(v(k))−1) We demonstrate in the same way that n v(k + 1) = v(k) + i=1 Û Û di (k)xi (k)(exp(ui (k)) − 1) − e(k)y (k)(exp(v(k)) − 1) According to a Taylor’s formula, there exist θi,1 , θi,2 , θ in ]0, 1[ such that exp(3ui (k)) − 1 = 3ui (k) exp(3θi,1 ui (k)), exp(2ui (k)) − 1 = 2ui (k) exp(2θi,2 ui (k)), and exp(v(k)) − 1 = v(k) exp(θv(k)). This implies: 148 Ä ui (k + 1) = ä Û Û Û ui (k) 1 − 3xi (k)3 exp(3θi,1 ui (k)) + 2α(k)xi (k)2 exp(2θi,2 ui (k)) − c(k)v(k)y (k) exp(θv(k)) So n v(k + 1) = Û v(k) [1 − e(k)y exp(θv(k))] + Ç å i=1 å Ç Û di (k)xi (k)ui (k) exp(θi ui ) 2 pi 3 q u 2 pi |ui (k + 1)| ≤| ui (k)|(1 + + 2α pi ) + cu |v(k)|q pi ˜ pi ˜ q ˜ Ç å3 Ç å2 pi q pi ≤ max 1 + 3p3 max(|v(k)|, |u1 (k)|, · · · , |un (k)|) + 2αu p2 , cu q i i pi ˜ pi ˜ q ˜ 3p3 i And for v: Ç å Ç n q ˜ pi |v(k + 1)| ≤ |v(k)||1 − e q ˜ |+ du pi |ui (k)| i q pi ˜ i=1 l Ç å q ˜ 1−eq ˜ q l ≤ Let’s call and assume λi = max 1 + d u pi i + i=1 3p3 i Ç å q ˜ δ = 1−eq ˜ q l n Ç pi pi ˜ + å3 n Ç pi pi ˜ + å max(|v(k)|, |u1 (k)|, · · · , |un (k)|) 2αu p2 i d u pi i i=1 Ç å pi pi ˜ Ç å pi pi ˜ å2 , cu q q <1 q ˜ <1 Let’s call ζ = max(λi , δ). We have ζ < 1. Then we get max(|ui (k + 1)|, |v(k + 1)|) ≤ ζmax(|ui (k)|, |v(k)|) So for k∗ k max(|ui (k)|, |v(k)|) ≤ ζ k−k∗ max(|ui (k∗)|, |v(k∗)|) Then by taking the limit we get: So limk→+∞ max(|ui (k)|, |v(k)|) = 0. lim ui (k) = 0, k→+∞ i = 1, 2, · · · , n lim v(k) = 0 k→+∞ Which leads to Û lim (y(k) − y (k)) = 0 k→+∞ ı lim (xi (k) − xi (k)) = 0, k→+∞ 149 i = 1, 2, · · · , n The positive solution of the system is stable. There is a stable positive periodic solution There exists a positive integer ω that is a common multiple of the period of all coefficients of system (1.3). Let’s notice Dn+1 = [˜1 , p1 ] × · · · × [˜n , pn ] × [˜, q]. p p q We consider the application φ that maps an initial state (x1 (0), · · · , xn (0), y(0)) ∈ Dn+1 to the solution of system (1.3) at generation ω. From theorem 1 and theorem 2, we demonstrated that the set Dn+1 is an invariant set of system (1.3). So φ maps the compact set Dn+1 to itself. Therefore, according to the Brouwer fixed point theorem, we know that the continuous application φ has a fixed point. Moreover the solution of (1.3) passing through this fixed point is periodic. From the theorem 3, we know that this positive periodic solution is stable. Therefore we demonstrated that our system (1.3) has one stable positive periodic solution. 150