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Dynamics of a stochastic delayed Harrison-type predation model: Effects of delay and stochastic components

  • * Corresponding author: Yun Kang

    * Corresponding author: Yun Kang
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  • This paper investigates the complex dynamics of a Harrison-type predator-prey model that incorporating: (1) A constant time delay in the functional response term of the predator growth equation; and (2) environmental noise in both prey and predator equations. We provide the rigorous results of our model including the dynamical behaviors of a positive solution and Hopf bifurcation. We also perform numerical simulations on the effects of delay or/and noise when the corresponding ODE model has an interior solution. Our theoretical and numerical results show that delay can either remain stability or destabilize the model; large noise could destabilize the model; and the combination of delay and noise could intensify the periodic instability of the model. Our results may provide us useful biological insights into population managements for prey-predator interaction models.

    Mathematics Subject Classification: Primary: 34C25, 34F05; Secondary: 92B05.


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  • Figure 1.  Phase portrait of model (2) and the parameters are taken as $c = 0.9, \, b = 0.7, \, d = 0.3, \, m = 0.1$. The horizontal axis is prey population $N$ and the vertical axis is predator population $P$. The red dotted curve is the $N$-isoline $cP = (1-N)(mP+1)$ and the yellow solid curve is the $P$-isoline $bN = d(mP+1)$. Both $E_0 = (0, 0)$ and $E_1 = (1, 0)$ are saddle points, $E^* = (0.46, 0.64)$ is locally asymptotically stable

    Figure 2.  The effects of the time delay $\tau$ on the dynamics of the DDE model (4) when $c = 0.9, \, b = 0.7, \, d = 0.3, \, m = 0.1$ which are the same as in Fig. 1. In the figures of time series, the red curve is the population of $N$ and the blue curve is the population of $P$

    Figure 3.  Time-series plots of model (3) without time-delay and only with different noises $\sigma_1, \, \sigma_2$, and other parametric values are $\tau = 0, \, c = 0.9, \, b = 0.7, \, d = 0.3, \, m = 0.1$

    Figure 4.  Time-series plots of the SDDE model (3) for different noise $\sigma_1, \, \sigma_2$ with time delay $\tau = 2.8 < \tau_0 = 3.46$, other parametric values are given as (16)

    Figure 5.  Time-series plots of the SDDE model (3) for different noise $\sigma_1, \, \sigma_2$ with $\tau = 3.9>\tau_0 = 3.46$, other parametric values are given as (16)

    Table 1.  The existence and stability of equilibria for model (2) where $N^* = \frac{b(m-c)+\sqrt{4bcdm+b^2(m-c)^2}}{2bm}, \, P^* = \frac{bN^*-d}{dm}$

    Equilibrium Existence Condition Stability Condition
    $(0, 0)$ Always exists Always saddle
    $(1, 0)$ Always exists Sink if $d\geq b$;
    Saddle if $d < b$
    $(N^*, P^*)$ $d < b$ Always sink
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