In this dissertation, we developed a new population dynamics model from probability theory and logistic model:\begin{eqnarray} u_t&=&(D(u)u_x))_x,\label{eq:0001} \\ D(u)&<&0, \quad \mbox{for} \quad u\in (0,\alpha);\quad D(u)>0 \quad \mbox{for} \quad u\in (\alpha,1).\label{eq:0002}\end{eqnarray} These equations describe population aggregation when the population density is small and diffusion when the population is large. When the net birth is also considered, the equation becomes:\begin{equation}\label{eq:0003} u_t=(D(u)u_x))_x+g(u).\end{equation}We assume that the net birth $g(u)$ satisfies\begin{equation}\label{eq:0004}g(0)=g(1)=0,\quad g(u)>0 \quad \forall u\in(0,1).\end{equation}Because of the singularity at point $u=\alpha$, we can not obtain the traveling wave solution easily by methods used for purely diffusion model (which means $D(u)\geq 0$ for all $u\in [0,1]$). In order to overcome this difficulty, we introduce a suitable definition of weak traveling wave solution and obtain the existence of the traveling wave solutions depending on the traveling wave speed $c$.\par Secondly, we consider some properties of the weak solution of Equation \eqref{eq:0001} with non-flux boundary conditions such as existence and nonexistence of the weak solution when $x\in \Omega\subseteq \mathbb{R}^1$ and the asymptotic behavior of the solution:\begin{equation}\label{eq:0005} \lim_{t\rightarrow+\infty}u(x,t)=\frac{1}{|\Omega|}\int_{\Omega}u(x,0)dx, \quad u(x,0)\geq \alpha.\end{equation}\par In the third part, we consider the original discrete model which has solution naturally under general initial condition. We first prove the discrete population density $0\leq u(j,t)\leq 1$ which satisfies our assumptions before deriving our new population model. Secondly, we obtain the asymptotic behavior of this discrete model when $u(j,0)\geq \alpha=1/2, j=0,1,2,\dots,N$ and $u(j,0)$ is monotone:\begin{equation}\label{eq:0006} \lim_{t\rightarrow +\infty}u(j,t)=\frac{1}{N-1}\sum_{j=1}^{N-1}u(j,0).\end{equation}When $N\leq4$ and $u(0,t)=u(N,t)=0$, we obtain the convergence results of the solutions as $t\rightarrow +\infty$ under very general initial conditions.One more very interesting question is: How does the forward region $Q_d^+(t)=\{(i,t)| u(i,t)\geq \alpha,i\in [1,2,3,\dots,N]\}$ and backward region $Q_d^-(t)=\{(i,t)| u(i,t)\leq \alpha,i\in [1,2,3,\dots,N]\}$ change as time progresses? We can prove under very general initial conditions:\begin{equation}\label{eq:0007} Q_d^+(t)\subseteq Q_d^+(t_1), \quad\forall t\leq t_1.\end{equation}\par In the last part, we consider the traveling wave solution for a cell-to-cell model with adhesion which is different to our first model. \begin{equation}\label{eq:0008} \rho_t= [ D(\rho)\rho_x]_x + g(\rho)\quad t\geq 0, \quad x\in \mathbb{R},\end{equation}with quadratic coefficient\begin{equation}\label{eq:0009} D(\rho) = 3\gamma \rho^2-4\gamma\rho+1,\end{equation}where $\rho(x,t)$ represent the cell density and $\gamma$ is the adhesive coefficient between cells. We obtain the following existence theorem:\\ $\mathbf{Theorem}$ Let $D(\rho)$ and $g(\rho)$ be given functions respectively satisfying \eqref{eq:0009} and \eqref{eq:0004} and $3/4\leq \gamma\leq 1$.There exists a value $c^*>0$, satisfying\begin{eqnarray*} &&\max\{D'(0)g(0), D'(\rho^{\sharp}(\gamma))g(\rho^{\sharp}(\gamma))\}\leq \frac{(c^*)^2}{4} \\ &&\leq \max\{\sup_{s\in(0, \rho^{\flat}(\gamma)]}\frac{D(s)g(s)}{s},\sup_{s\in(\rho^{\flat}(\gamma),1],\rho\neq\rho^{\sharp}(\gamma)}\frac{D(s)g(s)}{s-\rho^{\sharp}(\gamma)}\},\end{eqnarray*}such that Equation \eqref{eq:0008} has\par i) no weak traveling wave solution for $c c^*$.\par Where $(\rho^{\flat}(\gamma),\rho^{\sharp}(\gamma)) := (\frac{2\gamma-\sqrt{\gamma(4\gamma-3)}}{3\gamma},\frac{2\gamma+\sqrt{\gamma(4\gamma-3)}}{3\gamma})$.
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