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Existence and global stability of periodic solution for impulsive predator-prey model with diffusion and distributed delay. (English) Zbl 1219.34106

The authors study the following predator-prey model with delay, prey dispersal and impulsive harvesting
\[ \begin{cases} \dot x_1(t)=x_1(t)(a_1(t)-b_1(t)x_1^{\alpha}(t)-c(t)y^{\gamma}(t))+D_1(t)(x_2(t)-x_1(t)),\\ \dot x_2(t)=x_2(t)(a_2(t)-b_2(t)x_2^{\beta}(t))+D_2(t)(x_1(t)-x_2(t)),\\ \dot y(t)=y(t)\big(-d(t)+e(t)x_1^{\alpha}(t)-q(t)y^{\gamma}(t)-p(t)\int_{-\tau}^{0}K(s)y^{\gamma}(t+s)\,ds\big),\\ t\neq \tau_k, \quad k\in\mathbb Z_+,\\ \Delta x_1(\tau_k)=c_kx_1(\tau_k),\;\Delta x_2(\tau_k)=d_k x_2(\tau_k),\;\Delta y(\tau_k)=e_k y(\tau_k), \end{cases}\tag{1} \]
where \(x_1(t)\) and \(x_2(t)\) represent the densities of the prey species in patches 1 and 2 at time \(t\), respectively; \(y(t)\) represents the density of the predator at time \(t\). \(a_i, b_i\) \((i=1,2)\), \(c, d, e, q\) and \(p\) are continuous and positive periodic functions with period \(\omega\), \(\tau\) is a nonnegative constant. \(K(s)\geq 0\), \(s\in [-\tau,0]\), \(K\) is a piecewise continuous and normalized function such that \(\int_{-\tau}^{0}K(s)\,ds=1\). \(\alpha, \beta, \gamma\) are positive constants, \(-1<c_k\leq 0\), \(-1<d_k\leq 0\), \(-1<e_k\leq 0\) for \(k\in\mathbb Z_+\) \((\mathbb Z_+=\{1,2,\dots\})\), there exists \(q\) such that \(\tau_{k+q}=\tau_k+\omega\), \(c_{k+q}=c_k\), \(d_{k+q}=d_k\), \(e_{k+q}=e_k\).
By using the coincidence degree theorem, a set of easily verifiable sufficient conditions are obtained for the existence of at least one strictly positive periodic solution. By means of a suitable Lyapunov functional, uniqueness and global attractiveness of a positive periodic solution are established.

MSC:

34K60 Qualitative investigation and simulation of models involving functional-differential equations
92D25 Population dynamics (general)
34K13 Periodic solutions to functional-differential equations
47N20 Applications of operator theory to differential and integral equations
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