## The dynamics of an impulsive delay predator-prey model with stage structure and Beddington-type functional response.(English)Zbl 1218.34099

The authors consider the following stage-structured predator-prey model with impulsive effect
$\begin{cases} \dot x(t)=x(t)(a-bx(t))-\frac{cx(t)y_2(t)}{\alpha+x(t)+\beta y_2(t)}, &t\neq nT, \\ \dot y_1(t)=\lambda\frac{cx(t)y_2(t)}{\alpha+x(t)+\beta y_2(t)}-d_1y_1(t)-\lambda e^{-d_2\tau}\frac{cx(t-\tau)y_2(t-\tau)}{\alpha+x(t-\tau)+\beta y_2(t-\tau)}, &t\neq nT,\\ \dot y_2(t)=\lambda e^{-d_2\tau}\frac{cx(t-\tau)y_2(t-\tau)}{\alpha+x(t-\tau)+\beta y_2(t-\tau)}-d_2y_2(t)-ry_2^2(t), &t\neq nT,\\ x(t^+)=(1-p)x(t),\;y_1(t^+)=y_1(t)+\mu, \;y_2(t^+)=y_2(t), &t=nT, \end{cases}\tag{1}$
where $$x(t), y_1(t), y_2(t)$$ represent the densities of prey, immature predator and mature predator populations at time $$t$$, respectively. $$a$$ is the intrinsic growth rate of the prey, $$b$$ is the intra-specific competition rate of the prey, $$c$$ is the capturing rate of the predator, $$\alpha$$ is a saturation constant, $$\beta$$ scales the impact of the predator interference, $$\lambda$$ is the conversion coefficient, $$d_1, d_2$$ are the mortality rates of the immature and mature predator, respectively, $$r$$ is the intra-specific competition rate of the mature predator, $$\tau$$ represents a constant time to maturity for the immature predator, $$p$$ $$(0\leq p<1)$$ represents the partial impulsive harvest of the prey by catching or pesticide, $$\mu\geq 0$$ is the amount of immature predators released at fixed time $$t=nT$$.
By using the discrete dynamical system determined by the stroboscopic map, the existence and global attractivity of the predator-extinction periodic solution are obtained. Based on the theory of impulsive delay differential equations, the permanence of the system is studied.

### MSC:

 34K60 Qualitative investigation and simulation of models involving functional-differential equations 34K45 Functional-differential equations with impulses 34K25 Asymptotic theory of functional-differential equations 92D25 Population dynamics (general) 34K13 Periodic solutions to functional-differential equations 34K20 Stability theory of functional-differential equations

### Keywords:

attractivity; delay; impulsive effect; extinction; permanence
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### References:

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