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

  • *Corresponding author: Hirofumi Izuhara

    *Corresponding author: Hirofumi Izuhara 

In memory of the late Professor Masayasu Mimura

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  • 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:

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  • 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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