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Delay-induced instabilities of stationary solutions in a single species nonlocal hyperbolic-parabolic population model

  • * Corresponding author: Majid Bani-Yaghoub

    * Corresponding author: Majid Bani-Yaghoub 
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  • The present work investigates the effects of maturation and dispersal delays on dynamics of single species populations. Both delays have been incorporated in a single species nonlocal hyperbolic-parabolic population model, which admits traveling and stationary wave solutions. We reduce the model into various forms and obtain the corresponding analytical solutions. Analysis of the reduced models indicates that the dispersal delay can result in loss of monotonicity, where the solutions oscillate as they converge to a positive equilibrium. The stability analysis of the general model reveals that the maturation time delay admits a Hopf bifurcation threshold, which is expressed as a function of the dispersal delay. The numerical simulations of the general model suggest that the global stability of the stationary wave solutions is lost when the dispersal delay is increased from zero. In conclusion, population models with maturation and dispersal delays can give new insights into the complex dynamics of single species.

    Mathematics Subject Classification: 37N25, 35R10.


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  • Figure 1.  A schematic representation of the delayed HPRD model (26). Individuals can randomly move between infinitely many habitats on a straight line. Considering habitats centered at $ x_{i-1}, x_{i} $ and $ x_{i+1} $, the effects of birth function $ b(w) $, mortality rate $ d_{m}, $ survival rate $ \varepsilon $, dispersal delay $ r $, migration between habitats $ f_{\alpha}\ast b(w) $, and migration from other regions $ k_{i} $ are indicated in the diagram

    Figure 2.  Oscillation due to dispersal delay. (a) without dispersal delay (i.e. when r = 0), the solution of the reduced model (30) converges monotonically to the constant solution $ k/d_{m} = 2 $ as $ t \rightarrow \infty. $ (b) when $ r>0 $, the convergence occurs non-monotonically and the solution $ w(x, t) $ oscillates around $ k/d_{m} = 2. $ See the video clips "monotonic.gif" and "oscillatory.gif" in the supplementary document corresponding to panels (a) and (b) respectively

    Figure 3.  Graph of the Hopf bifurcation value $ \hat{\tau}(r) $ as an increasing function of the dispersal delay $ r $. See the text for the specific parameter values

    Figure 4.  The destabilizing effect of dispersal delay on the stationary wave pulse of the general model (26). Panels (a) - (d) show that the stability of the stationary pulse is lost as the value of dispersal delay $ r $ is increased from zero. The values of $ r $ are indicated in each panel. See the video clip "Pulse.gif" in the supplementary document animating the solutions in panels (b) - (d)

    Figure 5.  The destabilizing effect of dispersal delay on the stationary wavefront of the general model (26). Panels (a) - (d) show that the stability of the stationary wavefront is lost as the value of dispersal delay $ r $ is increased from zero. The values of $ r $ are indicated in each panel. See the video clip "Front.gif" in the supplementary document animating the solutions in panels (b) - (d) with respect to time $ t $

    Table 1.  A summary of the parameters and functions used in the general model (26)

    Symbol Description Dimension
    $ \tau $ maturation time delay $ [T] $
    $ r $ dispersal delay $ [T] $
    $ k $ migration rate (source) $ [T]^{-1} $
    $ D_m $ diffusion rate of the mature population $ [L]^{2}[T]^{-1} $
    $ D_I(a) $ diffusion rate of the immature population at age $ a $ $ [L]^{2}[T]^{-1} $
    $ d_I(a) $ mortality rate of the immature population at age $ a $ $ [T]^{-1} $
    $ d_m $ mortality rate of the mature population $ [T]^{-1} $
    $ \varepsilon $ survival rate of the mature population $ - $
    $ b(w) $ birth function (birth rate) $ - $
    $ f_{\alpha}(x) $ survival probability after traveling x units (Gaussian function) $ [L]^{-1} $
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    Table 2.  A summary of the description and outcomes of the reduced models (26)

    Model $ D_{I} $ $ D_{M} $ $ r $ $ \varepsilon $ $ \tau $ Model Outcomes Refs
    (28) 0 0 0 0 0 globally stable equilibrium at $ w* =k/d_m $ [45]
    (29) 0 0 + 0 0 oscillations about $ w* = k/d_m $ [15,22]
    (30) 0 + + 0 0 spacial oscillations about $ w* $ [45]
    (31) 0 0 0 + 0 survival-extinction, survival of the species [45]
    (32) 0 0 0 + + delay-induced, bifurcating solutions [15,22]
    (33) 0 + 0 + 0 target waves, stationary & traveling waves [43,18]
    (34) 0 + 0 + + periodic solution in the spatial domain [18]
    (36) + + 0 + + wave approximation, spatial survival [54,5,34]
    Model Descriptions: Equation (28): spatially homogeneous without delay or survival of immature population; Equation (29): spatially homogenous with dispersal delay; Equation (30): dispersal delay and diffusion are present; Equation (31): spatially homogeneous without any delay term; Equation (32): spatially homogeneous with maturation time delay; Equation (33): local reaction-diffusion model; Equation (34): delayed local reaction-diffusion; Equation (36): nonlocal delayed RD equation.
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