| Contents |
| We are concerned with size-structured population models with spacial diffusion.
Consider a biological population living in a habitat $\Omega \subset \mathbb{R}^{n}$ with smooth boundary $\partial \Omega$.
Let $p(s,t,x)$ be the population density of size $s\in [0,s_{\dagger})$ and position $x\in \Omega$ at time $t\in [0,T]$, where $s_{\dagger}\in (0,\infty)$ is the maximal size, $T\in (0,\infty)$ is a given time.
Let $\Omega_{T}:=(0,T)\times \Omega$, $Q:=(0,s_{\dagger})\times \Omega$, $\mathcal{Q}_{T} := (0,s_{\dagger})\times (0,T)\times \Omega$ and $\Sigma_{T} := (0,s_{\dagger})\times (0,T)\times \partial \Omega$.
We first consider the following linear model:
\begin{equation*}
\begin{aligned}
& \partial_{t}p + \partial_{s}(g(s,t)p) = \Delta p(s,t,x) -\mu (s,t,x) p(s,t,x) + f(s,t,x),
&& \text{in $\mathcal{Q}_{T}$}, \\
& g(0,t) p(0,t,x) = C(t,x)+\int_{0}^{s_{\dagger}} \beta (s,t,x) p(s,t,x) \, ds,
&& \text{in $\Omega_{T}$}, \\
& \frac{\partial p}{\partial \nu} (s,t,x) = 0,
&& \text{on $\Sigma_{T}$},\\
& p(s,0,x) = p_{0}(s,x),
&& \text{in $Q$}.
\end{aligned}
\end{equation*}
Here $g(s,t)$ represents the growth rate depending on individual's size $s$ and time $t$.
The functions $\mu(s,t,x)$ and $\beta(s,t,x)$ stand for the mortality and fertility rates, respectively.
The spacial diffusion is prescribed by Laplacian $\Delta$ and the Neumann boundary condition.
The functions $f(s,t,x)$ and $C(t,x)$ represent certain inflows of $s$-size and zero-size populations, respectively, from outside.
We introduce a notion of mild solution and establish the existence of a unique mild solution.
Next, we develop a nonlinear problem in which mortality and fertility rates are supposed to depend on the population density $P(x,t)= \int_{0}^{s_{\dagger}} p(s,t,x)\, ds$ in position $x$ at time $t$. |
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