This thesis entitled “Hunting cooperation among predators: A mathematical study of
ecological models” attempts to study the effect of hunting cooperation among predators in ecological systems. In nature, many predator species cooperate during hunting for a variety of reasons, such as (i) hunting and killing large prey, (ii) searching for food, (iii) increasing the ability to capture (or subdue) prey, (iv) attacking a herd of prey, (v) increasing vigilance and protection against other predators, (vi) preventing the theft of corpses by other hunters, (vii) reducing the distance of chasing, etc. Thus how predators enhance their biomass by group hunting, and as a result, how they impact prey biomass is a natural issue. Although, the study of cooperative behaviour during hunting is not new, in the mathematical modelling approach there are only a few studies. Thus the main objective of the study is to observe the dynamics of two-species predator-prey models in the presence of hunting cooperation among predators.
In Chapter 1, first, we briefly described the evolution of mathematical models in ecology with the emphasis of two-species interaction and different types of functional response. Then characterize the cooperative hunting phenomenon in an ecological context. Here, we mainly concern with the following ecological issues: (1) why predators cooperate during hunting? (2) example of different cooperative hunting species and their strategies, (3) how to incorporate cooperative hunting phenomenon in ecological models?
In Chapter 2, we consider a discrete-time predator-prey model with logistic type prey growth to study the impact of hunting cooperation. We investigate the basic analysis of the discrete model such as fixed points, stability, and bifurcation analysis, and explore that hunting cooperation has the potential to modify the well-known period-doubling route to chaos by reverse period-halving bifurcations and makes the system stable. Also, for an additional increase of the strength of hunting cooperation, the system exhibits chaotic oscillations via Neimark-Sacker bifurcation. However, very high hunting cooperation can be detrimental for the system and populations go to extinction. This is because of the overexploitation of the prey populations by their predators. The discrete system shows bistability behaviour between prey only fixed point and interior fixed point, and the basin of attraction of the interior fixed point increases with the strength of hunting cooperation. Moreover, hunting cooperation induces a strong demographic Allee effect in the discrete system, where predator populations persist due to cooperation during hunting and would go to extinction without hunting cooperation.
In Chapter 3, we extend the continuous version of the model studied in the previous chapter, by incorporating predator induce fear in the birth rate of the prey population. We observe that without hunting cooperation, the unique coexistence equilibrium point is globally asymptotically stable. However, an increase in the hunting cooperation induce fear may destabilize the system and produce periodic solution via Hopf bifurcation. We explore very rich dynamics such as both supercritical and subcritical Hopf bifurcations, Bogdanov-Takens bifurcation, backward bifurcation, and different types of bistabilities. This model also generates strong demographic Allee effect in predator species.
In Chapter 4, we study another type of model namely, the modified Leslie-Gower model with the same phenomena, hunting cooperation in predators and fear effect in prey. The main feature of Leslie-Gower model with compare to the Lotka-Volterra model is the logistic growth of predators, where the carrying capacity of the predator species is proportional to the prey biomass. We observe that the fear factor can stabilize the predator-prey system by excluding the existence of periodic solutions and makes the system more robust compared to hunting cooperation. This system also shows very rich dynamics such as Hopf bifurcation, Bogdanov-Takens bifurcation, and multi-type bistabilities.
Chapter 5 is devoted to exploring the impact of time delay during cooperative hunting in a predator-prey model. Cooperative hunting predators do not aggregate in a group instantly but individuals use different stages and strategies such as tactile, visual, vocal cues/signals, or a suitable combination of these to communicate with each other. It is indeed plausible to add some time delay representing the delay in forming a group and ready for attack. Generally, delay has a destabilizing effect on predator-prey dynamics, but in our model, delay has both stabilizing as well as destabilizing effects. Also, for an increase in the strength of the delay, then system dynamics switch multiple times and finally become chaotic. We see that depending on the threshold of time delay, the system may restore its original state or may go far away from its original state and unable to recollect its memory. We also observe different kinds of multistability behaviours, coexistence of multiple attractors, and interesting changes in the basins of attractions of the system. We infer that depending on the initial population size and the strength of cooperation delay, the populations can exhibit stable coexistence, oscillating coexistence, or extinction of the predator species.
Hunting cooperation has both stabilizing and destabilizing effects on the dynamics of the systems depending on the values of model parameters. It can mediate the survival of predators, where predators go extinct without cooperation. Mathematical models with hunting cooperation among predators may exhibit different types of bistabilities (node-node/node-cycle). This can be biologically interpreted as, depending on the initial population size, the populations can exhibit stable coexistence, oscillating coexistence, or extinction of the predator species (for the case of specialist predators). Predator-prey models with hunting cooperation among predators exhibit rich dynamical behaviours. We believe that, this research work will definitely enrich the existing knowledge about the impact of hunting cooperation in predators on various mathematical models in ecology.
