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Delay induced spatiotemporal patterns in a diffusive intraguild predation model with Beddington-DeAngelis functional response

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  • A diffusive intraguild predation model with delay and Beddington-DeAngelis functional response is considered. Dynamics including stability and Hopf bifurcation near the spatially homogeneous steady states are investigated in detail. Further, it is numerically demonstrated that delay can trigger the emergence of irregular spatial patterns including chaos. The impacts of diffusion and functional response on the model's dynamics are also numerically explored.

    Mathematics Subject Classification: Primary: 35B32, 35B36, 35K57; Secondary: 92D25, 92D40.


    \begin{equation} \\ \end{equation}
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  • Figure 1.  Numerical solutions of (4) with $\tau = 0.7<\tau^\ast\approx0.7895$ (only the $u_1$ component is plotted here): the positive spatially homogeneous steady state is locally stable.

    Figure 2.  Numerical solutions of (4) with $\tau = 1.2>\tau^\ast\approx0.7895$: a periodic solution bifurcates from the positive spatially homogeneous steady state $E^\ast$.

    Figure 3.  Numerical solutions of the temporal model (left) and numerical solutions of the spatiotemporal model (right) with $\tau = 1$, $(P_2)$ and $(IC_2).$ Here, for the spatiotemporal model (4), average population density for each species is plotted.

    Figure 4.  Numerical solutions of the temporal model (left) and numerical solutions of the spatiotemporal model (right) with $\tau = 1.5$, $(P_2)$ and $(IC_2').$ Here, periodic oscillations are observed for the temporal model and chaotic behavior is observed for the spatiotemporal model.

    Figure 5.  Snapshots of contour maps of the basal resource $u_1$ for the temporal model (left) and spatiotemporal model (right) at $t = 2000$ with $\tau = 1.5$, $(P_2)$ and $(IC_2^\prime).$

    Figure 6.  Snapshots of contour maps of the time evolution of the specie $u_1$ at $t = 200$, $500$, $1000$, $1200$, $1500$, $2500$ with $\tau = 1.5$ under $(P_2)$ and $(IC_2^\prime)$.

    Figure 7.  Snapshots of contour maps of the time evolution of the basal resource $u_1$ with different values of $\tau$ at time $t = 1500$ under $(P_2)$ and $(IC_2^\prime)$. $(\mathrm{ⅰ})\,\tau = 0.86; (\mathrm{ⅱ})\,\tau = 1; (\mathrm{ⅲ})\,\tau = 1.2; (\mathrm{ⅳ})\,\tau = 1.4; (\mathrm{ⅴ})\,\tau = 1.6; (\mathrm{ⅵ})\,\tau = 1.9.$

    Figure 8.  Snapshots of contour maps of the basal resource $u_1$ at time $t = 1500$ with different diffusion coefficients, $\tau = 1.5$, under $(P_2)$ and $(IC_2^\prime)$.

    Figure 9.  Snapshots of contour maps of the time evolution of the basal resource $u_1$ with different values of $b$ and parameter values $\alpha = 0.7, \beta = 0.9, \beta_1 = 1.95, \beta_2 = 1.85, \gamma_1 = 0.2, \gamma_2 = 0.8, c = 5$ at times $t = 1500$ and $\tau = 1.5$ under$(IC_2^\prime)$.

    Figure 10.  10. Snapshots of contour maps of the time evolution of the basal resource $u_1$ with different values of $c$ and parameter values $\alpha = 0.7, \beta = 0.9, \beta_1 = 1.95, \beta_2 = 1.85, \gamma_1 = 0.2, \gamma_2 = 0.8, b = 0.25$ at times $t = 1500$ and $\tau = 1.5$ under$(IC_2^\prime)$.

    Table 1.  Parameters definitions in model (3) and their units, where [resource] indicates basal resource density, [IG prey] indicates IG prey density, and [IG predator] indicates IG predator density

    Symbol Parameter Definition Units
    $r$ Basal resource intrinsic growth rate [time]$^{-1}$
    $K$ Basal resource carrying capacity [Basal resource density]
    $c_1$ Predation rate of IG prey on resource [IG prey]$^{-1}$ [time]$^{-1}$
    $c_2$ Predation rate of IG predator on resource [IG predator]$^{-1}$[time]$^{-1}$
    $c_3$ Predation rate of IG predaotr on IG prey [IG preys][IG predator]$^{-1}$
    $e_1$ Conversion rate from resource to IG prey [IG preys][resource]$^{-1}$
    $e_2$ Conversion rate from resource to IG predator [IG predators][resource]$^{-1}$
    $e_3$ Conversion rate from IG prey to IG predator [IG predators][IG prey]$^{-1}$
    $a_1$ [Half saturation constant]$^{-1}$ [resource]$^{-1}$
    $a_2$ [Half saturation constant]$^{-1}$ [IG predator]$^{-1}$
    $m_1$ Mortality rate of IG prey [time]$^{-1}$
    $m_2$ Mortality rate of IG predator [time]$^{-1}$
    ${\widetilde d}_1$ Diffusion coefficient of resource [length]$^2$[time]$^{-1}$
    ${\widetilde d}_2$ Diffusion coefficient of IG prey [length]$^2$[time]$^{-1}$
    ${\widetilde d}_3$ Diffusion coefficient of IG predatior [length]$^2$[time]$^{-1}$
    $L$ The size of spatial domain $\Omega$ [length]
     | Show Table
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