ABSTRACT
In the current paper, we consider a predator-prey model where the predator is modeled as a generalist using a modified Leslie-Gower scheme, and the prey exhibits group defense via a generalized response. We show that the model could exhibit finite-time blow-up, contrary to the current literature [Patra et al., Eur. Phys. J. Plus 137(1), 28 (2022)]. We also propose a new concept via which the predator population blows up in finite time, while the prey population quenches in finite time; that is, the time derivative of the solution to the prey equation will grow to infinitely large values in certain norms, at a finite time, while the solution itself remains bounded. The blow-up and quenching times are proved to be one and the same. Our analysis is complemented by numerical findings. This includes a numerical description of the basin of attraction for large data blow-up solutions, as well as several rich bifurcations leading to multiple limit cycles, both in co-dimension one and two. The group defense exponent p is seen to significantly affect the basin of attraction. Last, we posit a delayed version of the model with globally existing solutions for any initial data. Both the ordinary differential equation model and the spatially explicit partial differential equation models are explored.
ABSTRACT
Mutual interference and prey refuge are important drivers of predator-prey dynamics. The "exponent" or degree of mutual interference has been under much debate in theoretical ecology. In the present work, we investigate the interplay of the mutual interference exponent, and prey refuge, on the behavior of a predator-prey model with a generalized Holling type functional response - considering in particular the "non-smooth" case. This model can also be used to model an infectious disease where a susceptible population, moves to an infected class, after being infected by the disease. We investigate dynamical properties of the system and derive conditions for the occurrence of saddle-node, transcritical and Hopf-bifurcations. A sufficient condition for finite time extinction of the prey species has also been derived. In addition, we investigate the effect of a prey refuge on the population dynamics of the model and derive conditions such that the prey refuge would yield persistence of the population. We provide additional verification of our analytical results via numerical simulations. Our findings are in accordance with classical experimental results in ecology (Gause, 1934), that show that extinction of predator and prey populations is possible in a finite time period - but that bringing in refuge can effectively yield persistence.
