Article Contents
Article Contents

# Cross-diffusion predator–prey model derived from the dichotomy between two behavioral predator states

• *Corresponding author: Hirofumi Izuhara

In memory of the late Professor Masayasu Mimura

• Cross-diffusion may be an important driving force of pattern formation in population models. Recently, a relation between cross-diffusion and reaction-diffusion systems has been revealed from the mathematical modeling point of view. In this paper, we derive a predator–prey model with cross-diffusion from a simple reaction-diffusion system with two behavioral states in the predator population and examine whether cross-diffusion can induce spatial patterns in predator–prey models. We assume that the predators have identical behavioral characteristics except for their mobility and searching activity for preys: we consider two states, namely less mobile predators searching for preys more actively than mobile predators. Our analysis shows that cross-diffusion derived in this situation can induce spatial patterns if the prey-density-dependent conversion rate from less mobile state to mobile one increases more rapidly than that from mobile to less mobile at high prey density.

Mathematics Subject Classification: Primary: 92D25, 35B36; Secondary: 35K57, 35B32, 35K20.

 Citation:

• Figure 1.  Graphs of the functions $h(N) = \alpha N$ and $k(N) = \beta e^{\gamma N}$, where the parameter values are $\alpha = 5.0$, $\beta = 0.1$ and $\gamma = 4.0$

Figure 2.  The graphs of the left- (solid curve) and right-hand sides (dashed curve) of the inequality (21) against $N^*$. For example, (21) is satisfied with $N^* = 1$. The parameters are $K = 4$, $\alpha = 5$, $\beta = 0.1$, $\gamma = 4$

Figure 3.  The graphs of $\det M$ against $\lambda_k$. The three graphs from bottom to top correspond to the cases where $d_N = 0.05$ (green), $0.1$ (blue), and $0.15$ (red), respectively. The other parameters are fixed at $K = 4$, $\alpha = 5$, $\beta = 0.1$, $\gamma = 4$, $s = 1$, $a_1 = a_2 = 1$, $c_1 = c_2 = 1$, $r = 1$, $d_1 = 1$, $d_2 = 0$. Note that the parameters yield $N^* = 1$ and the parameters $K$, $\alpha$, $\beta$, and $\gamma$ are the same as in Figure 2

Figure 4.  Neutral stability curves satisfying $\det M = 0$ in $(d_N, d_1)$-plane. The gray region means that the positive constant solution $(N^*, P^*)$ is stable. On each curve, the indicated $n$-Fourier cosine mode has at least one zero eigenvalue. The parameter values are fixed as (22)

Figure 5.  Bifurcation diagram for $d_1 = 1.0$. The parameter values are fixed as (22). The horizontal axis is $d_N$ and the vertical axis is the boundary value of stationary solutions $N(x)$ at $x = 0$. The solid and the dashed curves respectively denote stable and unstable branches. The symbol $\square$ indicates a pitchfork bifurcation point. Two solution profiles for $d_N = 0.1$ and $d_N = 0.06$ are displayed in the figure, where the red and the blue curves are respectively $N$ and $P$

Figure 6.  Bifurcation diagram for $d_1 = 3.0$. The parameter values are fixed as (22). The horizontal axis is $d_N$ and the vertical axis is the boundary value of stationary solutions $N(x)$ at $x = 0$. Two solution profiles for $d_N = 0.25$ and $d_N = 0.1$ are displayed

Figure 7.  Bifurcation diagram for $d_1 = 3.0$ when $d_N$ varies. (left) The vertical axis is $\|N\|_{L^1(0, \pi)}$. (right) The vertical axis is $\|P\|_{L^1(0, \pi)}$

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