A dynamic analysis of a prey – predator – superpredator system: a family of equilibria and its destruction

The paper investigates the dynamics of a finite-dimensional model describing the interaction of three populations: prey $x(t)$, its consuming predator $y(t)$, and a superpredator $z(t)$ that feeds on both species. Mathematically, the problem is formulated as a system of nonlinear first-order differential equations with the following right-hand side: $[x(1-x)-(y+z)g;$ $\eta_1yg-d_1f-\mu_1y;$ $\eta_2zg+d_2f-\mu_2z]$, where $\eta_j, d_j, \mu_j (j=1,2)$ are positive coefficients. The considered model belongs to the class of cosymmetric dynamical systems under the Lotka – Volterra functional response $g=x$, $f=yz$, and two parameter constraints: $\mu_2=d_2(1+\frac{\mu_1}{d_1})$, $\eta_2=d_2(1+\frac{\eta_1}{d_1})$. In this case, a family of equilibria is being of a straight line in phase space. We have analyzed the stability of the equilibria from the family and isolated equilibria. Maps of stationary solutions and limit cycles have been constructed. The breakdown of the family is studied by violating the cosymmetry conditions and using the Holling model $g(x)=\frac{x}{1+b_1x}$ and the Beddington–DeAngelis model $f(y,z)=\frac{yz}{1+b_2y+b_3z}$. To achieve this, the apparatus of Yudovich's theory of cosymmetry is applied, including the computation of cosymmetric defects and selective functions. Through numerical experimentation, invasive scenarios have been analyzed, encompassing the introduction of a superpredator into the predator-prey system, the elimination of the predator, or the superpredator.