A predator--prey model with inverse trophic relation and time delay (Q2572088)

The phenomenon that a prey can consume immature predator is called inverse trophic relation. In this paper, the authors propose the following prey-predator model, \[ \begin{aligned} x'(t) = & x(t)[r_1-a_{11}x(t)+\alpha_1 y(t)-a_{13}Y(t)]\,,\\ y'(t) =& -[r_2+\alpha_2 x(t)]y(t)+a_{31}x(t)Y(t) \\ & -a_{31}x(t-\tau)Y(t-\tau)e^{\int_{t-\tau}^t [-r_2-\alpha_2x(s)]\text{ d}s}\,, \\ Y'(t)=&-r_3Y^2(t)+a_{31}x(t-\tau)Y(t-\tau)e^{\int_{t-\tau}^t [-r_2-\alpha_2x(s)]\text{ d}s}\,, \end{aligned} \tag{\(*\)} \] to describe the inverse trophic relation. This model (*) is based on those studied by \textit{W. G. Aiello} and \textit{H. I. Freedman} [Math. Biosci. 101, 139--153 (1990; Zbl 0719.92017)] and the authors [Can. Appl. Math. Q. 11, 293--302 (2003; Zbl 1087.34551)]. Here, \(x\) is the density of prey, and \(y\) and \(Y\) denote the densities of the immature and mature predator populations, respectively. Though it is unrealistic from a biological point of view, as a first step to find the global bifurcation caused by the effect of an inverse trophic relation for (*), the case where \(\alpha_1=0\) and \(\alpha_2>0\) is considered in this paper. First, the following result is established by using the method of contracting rectangles. Theorem 1. Suppose \(\alpha_1=0\). Then, system (*) has a unique interior rquilibrium to which all the solutions tend as \(t\to \infty\) if \(a_{11}r_3>a_{13}a_{31}e^{-r_2\tau}\) holds. Then, by analyzing the characteristic equation of the linearized system of (*) about the interior equilibrium, the authors further proved the following result. Theorem 2. Suppose \(\alpha_1=0\) and that \(\tau_0\) is a positive value determined by \(a_{11}r_3=a_{13}a_{31}e^{-r_2\tau_0}\). Then, the interior equilibrium of (*) is globally asymptotically stable for all \(\tau>\tau_0\) if \(3a_{11}r_3\geq a_{13}a_{31}\) holds. Theorem 2 is very interesting because the local stability holds for \(\tau\) bigger than a critical value, which is different from many existing results in the literature. Moreover, Theorem 1 and Theorem 2 show that \(\alpha_2\) is not a \textit{destabilizer} for global properties of the interior equilibrium under the conditions that ensure global attractivity or global asymptotic stability in the case \(\alpha_1=\alpha_2=0\) (see the author's paper cited).