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Bifurcation and spatiotemporal patterns in a Bazykin predator-prey model with self and cross diffusion and Beddington-DeAngelis response

Dedicated to Professor Stephen Cantrell on the occasion of his 60th birthday.

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  • In this article, the clasical Bazykin model is modifed with Bedding–ton–DeAngelis functional response, subject to self and cross-diffusion, in order to study the spatial dynamics of the model.We perform a detailed stability and Hopf bifurcation analysis of the spatial model system and determine the direction of Hopf bifurcation and stability of the bifurcating periodic solutions. We present some numerical simulations of time evolution of patterns to show the important role played by self and cross-diffusion as well as other parameters leading to complex patterns in the plane.

    Mathematics Subject Classification: Primary:35B32, 35B35, 35B36;Secondary:35Q92.


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  • Figure 1.  Relation between $H(k^{2})$ and $ \Re( \lambda ) $ for $ \det( \Phi_w(w^{*}) ) <0$. (a) $H(k^{2})$ (b) $ \Re( \lambda ) $

    Figure 2.  Relation between $H(k^{2})$ (a) and $ \Re( \lambda ) $ (b) for $ \det( \Phi_w(w^{*}) ) >0$

    Figure 3.  Functions $s=0$ (solid line) and $g=0$ (dashed line)

    Figure 4.  Phase plane of local system for $ \beta=1.1 $

    Figure 5.  Prey and predator densities for $\alpha_{12}=0.0001$. a) $u$ with $t_{max}=0$, b) $v$ with $t_{max}=0$, c) $u$ with $t_{max}=50$, d) $v$ with $t_{max}=50$, e) $u$ with $t_{max}=200$, f) $v$ with $t_{max}=200$

    Figure 6.  Prey and predator densities for example 2, with $\alpha_{12}=0.0001$ and $t_{max}=400$. a) $u$ density b) $v$ density. Both figures show a pattern called holes

    Figure 7.  Prey and predator densities for example 3, with $\alpha_{12}=0.0001$ and $t_{max}=1000$. a) $u$ density b) $v$ density. Both figures show a pattern called stripes

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