Ecoepidemic predator–prey model with feeding satiation, prey herd behavior and abandoned infected prey

Abstract

MSC: Keywords:Herd behavior Disease transmission Ecoepidemics System collapse Local and global bifurcations a b s t r a c tIn this paper we analyse a predator-prey model where the prey population shows group defense and the prey individuals are affected by a transmissible disease. The resulting model is of the RosenzweigMacArthur predator-prey type with an SI (susceptible-infected) disease in the prey. Modeling prey group defense leads to a square root dependence in the Holling type II functional for the predator-prey interaction term. The system dynamics is investigated using simulations, classical existence and asymptotic stability analysis and numerical bifurcation analysis. A number of bifurcations, such as transcritical and Hopf bifurcations which occur commonly in predator-prey systems will be found. Because of the square root interaction term there is non-uniqueness of the solution and a singularity where the prey population goes extinct in a finite time. This results in a collapse initiated by extinction of the healthy or susceptible prey and thereafter the other population(s). When also a positive attractor exists this leads to bistability similar to what is found in predator-prey models with a strong Allee effect. For the two-dimensional disease-free (i.e. the purely demographic) system the region in the parameter space where bistability occurs is marked by a global bifurcation. At this bifurcation a heteroclinic connection exists between saddle prey-only equilibrium points where a stable limit cycle together with its basin of attraction, are destructed. In a companion paper (Gimmelli et al., 2015) the same model was formulated and analysed in which the disease was not in the prey but in the predator. There we also observed this phenomenon. Here we extend its analysis using a phase portrait analysis. For the three-dimensional ecoepidemic predator-prey system where the prey is affected by the disease, also tangent bifurcations including a cusp bifurcation and a torus bifurcation of limit cycles occur. This leads to new complex dynamics. Continuation by varying one parameter of the emerging quasi-periodic dynamics from a torus bifurcation can lead to its destruction by a collision with a saddle-cycle. Under other conditions the quasi-periodic dynamics changes gradually in a trajectory that lands on a boundary point where the prey go extinct in finite time after which a total collapse of the three-dimensional system occurs.

A mathematical model for bovine tuberculosis among buffaloes and lions in the Kruger National Park

“…In Venturino and Petrovskii, this has been investigated, showing that trajectories lying in a narrow stripe may well end up on the prey axis in finite time, and from there, they move toward the origin, with ecosystem collapse. This has been further investigated in other works,() showing that it entails a wealth of bifurcation phenomena. For a generalization to an arbitrary power function instead of the use of the square root, see Bulai and Venturino …”

“…However, a particular phenomenon has been observed in similar models in these conditions. () The right‐hand side of the system is not Lipschitz continuous because of the presence of the square root in every component, so that the uniqueness theorem does not hold. In Venturino and Petrovskii, this has been investigated, showing that trajectories lying in a narrow stripe may well end up on the prey axis in finite time, and from there, they move toward the origin, with ecosystem collapse.…”

Consumer-resource coexistence as a means of reducing infectious disease

“…In the presence of a consumer and diseased resource, the overall system can show different complex dynamical behaviours, such as bistability, quasi-periodicity and chaos [10,32]. Here we use the classic Rosenzweig-MacArthur model approach for consumerresource dynamics which incorporates reasonable biology in the form of logistic resource growth and a saturating functional response of resource consumption, and can produce steady-state or cyclical dynamics [28].…”

Prey herd behavior modeled by a generic non-differentiable functional response

“…Nevertheless, an important issue that needs careful analysis is the behavior near the origin in the predatorprey model because the system cannot be linearized as with the square root functional response [11]. In addition, in [37], a numerical bifurcation analysis is presented that shows that there is non-uniqueness of the solution and a singularity where the prey population goes extinct in a finite time.…”