| Abstract: |
| The dynamical behaviour of a size-structured predator-prey model with nonlinear growth rates is studied. In the model prey growth rate decreases with prey population density, while predator growth rate depends on predation.
In its most general case the model is described by the following system of quasi-linear equations
where $u(t, s)$ and $v(t, s)$, for $t\ge 0$ and $s\in (0, 1)$, denote the prey and predator size-density, respectively, and, by appropriate scaling, 0 denotes the size at birth and 1 the maximal size:
\[
\left\{\setlength\arraycolsep{0.1em}\begin{array}{rlll} \displaystyle
\frac{\partial}{\partial t} u(t, s) + \frac{\partial}{\partial s}\left[g_1(U(t), s) u(t, s)\right]&=& -\left[d_1(s)+(Hv(t, \cdot))(s)\right]u(t, s), \ [4mm]
\displaystyle\frac{\partial}{\partial t} v(t, s) +
\frac{\partial}{\partial s} \left[g_2\left( (Ku(t,\cdot))(s), s\right)v(t,s)\right]&=& -d_2(s)v(t, s) ,\[2mm]
g_1(U(t), 0)u(t,0)&=&\displaystyle\int_0^{\bar s} \beta_1(s) u(t, s)\,ds, \[3mm]
g_2\left((Ku(t,\cdot) )(0), 0\right)v(t,0)&=&\displaystyle\int_0^{\bar s} \beta_2(s) v(t, s)\,ds,\[2mm]
\end{array}
\right.
\]
where $H, K\colon L^1((0, 1), \mathbf{R})\to L^1((0, 1), \mathbf{R})$ are
\[
(H\phi)(s)=\int_0^{1} k(z, s)\phi(z)\,dz,\qquad (K\phi)(s)=\int_0^{1} z k(s, z)\phi(z)\, dz,\quad s\in (0, 1),
\]
for $k\in L^1((0, 1)^2, \mathbf{R})$ %(in other words, $H$ is the adjoint of $K$),
and
\[
U(t)=\int_0^{1} u(t, s)\,ds,\qquad t\ge0.
\]
Existence and uniqueness of solutions are proved under weak conditions. A threshold is established for the stability of the predator-free equilibrium and the existence of a positive equilbrium. When restricted only to the prey, the model is a special case of the one studied by Farkas and Hagen (2007); in this case it is shown that the positive equilibrium may undergo Hopf bifurcation, a novel feature in this class of models. Finally, numerical exploration of the bifurcations is perfomed through pseudospectral collocation methods. |
|