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

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