7. Discussion
In predator-prey interaction, predation is considered to be the main force that promotes coexistence of competing species by reducing the strength of competition [6]. If the predator chooses strongest competitor species, mostly then it relives competition pressure on other species, thereby allowing coexistence of multiple species. Recent field experiments showed that predators can induce a non-consumptive effect on their prey, for example fear [23]. Due to predation fear, prey can adopt defensive strategies that disrupt coexistence [20].To address fears induced coexistence on competing species, we developed a mathematical model of two competing prey species and one predator where predator, not only kill both the prey but also shows non-consumptive effect upon them. Our system also includes intraspecific competition within the predator population. Takeuchi and Adachi [24] addresses an ecological system with the same type of species, but no fear effect, nor intraspecific competition within the predator populations obtaining coexistence results. The proposed model is biologically meaningful in the sense that any positive solution initiating in the positive orthant remains both non-negative and bounded.
Mathematical analysis of the model established that the system cannot collapse for any parameter value as the origin is always unstable. If the second prey has low intrinsic growth rate and the predator has a high death rate then the predator cannot prevent the first prey and tends to its carrying capacity;\(E_{1}\) is an attractor whereas the opposite hold if the first prey has low intrinsic growth rate. If the intraspecific competition if stronger than the interspecific competition and the predator has a high death rate then both the prey can coexist at \(E_{12}\) while predator population goes to extinction due to large death rate. The first prey and the predator can coexist at\(E_{13}\) when the second prey has moderate intrinsic growth rate. Again the second prey and the predator can coexist at \(E_{23}\ \)as long as the intrinsic growth rate remains below a certain threshold value. Using invasion analysis, we derived criterion for uniform persistence of our model system that ensures the existence of positive (coexistence) equilibrium point. Local stability of the coexistence equilibrium point is possible if the ratio of intake capacity by the predator lie within an interval. The existence of Hopf bifurcation is shown by considering the level of fear as bifurcation parameter. The nature of limit cycle emerging through a Hopf bifurcation is predicted by calculating the coefficient of curvature of the limit cycle. If the intraspecific competition of the first prey is less than that of second prey then supercritical limit cycle appears. In this paper we have not considered intraspecific competitive rate \(h\) as a bifurcation parameter. But one obtains the bifurcation result for taking \(h\) as bifurcation parameter. When most of the predators are involved in intraspecific competition, stable coexistence increases (see Fig. 1e).
The novelty of our work is the inclusion of fear effect and intraspecific competition within the predator populations which are not considered in [24]. This investigation generalizes the existing knowledge of fear effect of predator on single prey species [16-18, 27, 28, 30]. As high level of fear can destroy coexistence that agrees with [20]still coexistence of predator and competing prey is possible with the increase of intraspecific competition within the predator population. Our theoretical observations will be helpful to verify some experimental data set of two competing prey and one predator system.
It may also be worthwhile to see how the other response function rather than Holling type I affects the dynamics of the system. From experimental observation, we have considered the fear effect on reproduction term of prey population still it is reasonable to see the fear effect on intraspecific, interspecific competition or death rate of prey populations.
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