Periodic solution of a Lotka-Volterra predator-prey model with dispersion and time delays.
The paper contains an investigation of the following delay differential system \[ \begin{aligned} \frac{dx_1}{dt}&= x_1(r_1-a_{11}x_1-a_{13}y)+D_1(x_2-x_1),\\ \frac{dx_2}{dt}&= x_2(r_2-a_{22}x_2)+D_2(x_1-x_2),\\ \frac{dy}{dt}&= y(-r_3+a_{31}x_1(t-\tau _1)-a_{33}y(t-\tau _2)), \end{aligned} \] with typical initial conditions. The functions \(r_i\), \(a_{ij}\), \(D_k\) are continuous, positive and periodic with period \(w\), and \(\tau _i\) are nonnegative constants. The system describes the predator-prey interactions in patchy environments: \(x_1\) (prey), \(y\) (predator) denote the densities of species in patch \(1\), and \(x_2\) (prey) denotes the density of species in patch \(2\); \(r_k\) and \(D_k\) represent the intrinsic growth rate of the prey at patch \(k\), and the dispersion rate of prey species, respectively. The assumption of periodicity is a way of incorporating into the model the periodicity of the environment (seasonal effects of weather, food supplies, mating habits), while the delays \(\tau _1\), \(\tau _2\) are due to gestation of predators and to (negative) feedback of predator crowding, respectively. Denote \(\overline{f}=w^{-1}\int_0^wf(t)\,dt\), \(f^L= \min\{f(t):t\in [0,\omega ]\}\), \(f^M=\max\{f(t):t\in [0,\omega ]\}\). Then, the conditions (H2) \(a_{31}^L\overline{(r_1-D_1)}- \overline{r_3}a_{11}^M> 0\) and (H3) \(\overline{(r_2-D_2)}>0\) imply the existence of at least one positive \(w\)-periodic solution (Theorem 2.1) of the system. Two technical conditions (H4), (H5) are added to the hypotheses in order to ensure the uniqueness and global asymptotic stability of this solution (Theorem 4.1). The main result is established by means of the Gaines-Mawhin continuation theorem of coincidence degree theory. The computations are ingeniously performed and the entire paper is well-written and extremely well-documented. However, several errors (p. 543) have occured and propagated to the formula of a priori bound \(M\) (p. 547); the corrected estimates are: \[ \begin{aligned} \int_0^w\left| u_1^{\prime }(t)\right| \,dt &\leq 2\int_0^wa_{11}(t)e^{u_1(t)}\,dt+ 2\int_0^wa_{13}(t)e^{u_3(t)}\,dt+ \omega (r_1-D_1)^M; \tag{1}\\ \int_0^w\left| u_2^{\prime }(t)\right| \,dt &\leq 2\int_0^wa_{22}(t)e^{u_2(t)}\,dt+ \omega (r_2-D_2)^M; \tag{2}\\ \int_0^wa_{11}(t)e^{2u_1(t)}\,dt &< \int_0^w\left| r_1(t)-D_1(t)\right| e^{u_1(t)}\,dt+ \int_0^wD_1(t)e^{u_2(t)}\,dt; \tag{3}\\ \int_0^wa_{22}(t)e^{2u_2(t)}\,dt &< \int_0^w\left| r_2(t)-D_2(t)\right| e^{u_2(t)}\,dt+ \int_0^wD_2(t)e^{u_1(t)}\,dt. \tag{4} \end{aligned} \]
scientific article; zbMATH DE number 2019685