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December 2018, 15(6): 1401-1423. doi: 10.3934/mbe.2018064
Dynamics of a stochastic delayed Harrison-type predation model: Effects of delay and stochastic components
Feng Rao 1, , Carlos Castillo-Chavez 2,3, and Yun Kang 2,3,,
School of Physical and Mathematical Sciences, Nanjing Tech University, Nanjing, Jiangsu 211816, China
Mathematical, Computational and Modeling Sciences Center, Arizona State University, Tempe, AZ 85287-1904, USA
Sciences and Mathematics Faculty, College of Integrative Sciences and Arts, Arizona State University, Mesa, AZ 85212, USA
* Corresponding author: Yun Kang
Received December 10, 2017 Accepted June 10, 2018 Published September 2018
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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.
Keywords: Prey-predator model, time delay, Hopf bifurcation, stochastic perturbations, stability.
Mathematics Subject Classification: Primary: 34C25, 34F05; Secondary: 92B05.
Citation: Feng Rao, Carlos Castillo-Chavez, Yun Kang. Dynamics of a stochastic delayed Harrison-type predation model: Effects of delay and stochastic components. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1401-1423. doi: 10.3934/mbe.2018064
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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
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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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Feng Rao Carlos Castillo-Chavez Yun Kang
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