Dynamic response of a system of interactive species influenced by fear and Allee consequences

Abstract

The present pursuit is focused on the influence of both the facets of fear and Allee on the dynamic feedback of the model system of interacting species. Apart from the consideration of intra-specific competition of both species to make more dynamic intricacy, the ratio-dependent functional response is taken into account for the ecological compatibility of the system. The usage of the additive Allee effect helps one to estimate the impact on the interactions of predator and prey species. The particular circumstances for the extinction of both species are discussed based on the establishment of fundamental mathematical concepts necessary for the system. The existence of ecologically significant equilibria, including their stability, is explored. In spite of having a ratio-dependent functional response, the response of the system in proximity to the origin is examined by pursuing a particular method duly modified. The mandatory analytical parametric conditions for the stability criteria of the system are validated numerically through the exhibition of results obtained. The main finding of this research is that in a system of interacting species, fear and Allee may produce codimension 1 and 2 bifurcation structures. For parameter values within the bifurcating domain, additional theoretical dynamics are also formed and are clearly explored in the present research. The system experiences all possible local and global bifurcations including Hopf, saddle-node, Bautin, Bogdanov–Takens, and homoclinic subject to the combined influence of fear and Allee factors. The first Lyapunov number is made use of to analyse the stability of the Hopf-bifurcating limit cycle. The impact of fear and Allee effect around the coexistence equilibrium of the system is not ruled out, however, to examine from the present system. The sensitivity analysis of the model parameters with respect to fixed coexistence is also performed comprehensively. The numerical simulations are carried out finally for the purpose of validation of the present theoretical outcomes with supportive numerical counterparts and its application in the realm of ecology in the future.

Appendices

Appendix

A Basic model analysis

This section deals with the basic preliminaries of the proposed model system (2.3), such as positivity and uniqueness, dissipativeness, uniformly boundedness, and permanence, which ensure that the model system is well-posed. Additionally, we have discussed the extinction scenario for both species involved in the system (2.3).

Theorem A.1

All solutions (u(t), v(t)) of the model system (2.3) exist uniquely and all of them are positive for all $t \ge 0$.

Theorem A.2

The solutions of the proposed model system (2.3) satisfy the following;

$$\begin{aligned} \limsup _{t \rightarrow \infty } u(t) \le (\alpha -\beta ), \; \limsup _{t \rightarrow \infty } v(t) \le \frac{\epsilon -\delta }{\sigma }, \end{aligned}$$

(A.1)

provided $\alpha > \beta$ and $\epsilon > \delta$.

Remark A.3

Thus, from the inequalities (A.1) one may conclude that the proposed model system (2.3) is dissipative if the conditions $\alpha > \beta$ and $\epsilon > \delta$ hold good.

Theorem A.4

All solutions of the model system (2.3) along with their non-negative initial conditions, will ultimately lie within the region S defined by

$$\begin{aligned} S=\bigg \{(u, v) \in {\mathbb {R}}^{2}_{+}: 0 \le u(t)+v(t) \le W^{*}\bigg \} \end{aligned}$$

for all $t \ge 0$, where $W^{*}$ is specified in the proof section.

Theorem A.5

The proposed model system (2.3) is uniformly persistent if

$$\begin{aligned} \frac{\sigma \alpha }{\sigma + k(\epsilon -\delta )} > \beta +\frac{\gamma }{\rho }+\frac{\epsilon }{\eta }. \end{aligned}$$

Remark A.6

A model system is said to be permanent if it is dissipative as well as uniformly persistent [84]. Here, we have observed that the proposed model system (2.3) is dissipative through Theorem A.2 and uniformly persistent through Theorem A.5. Thus, the proposed model system (2.3) is permanent.

Theorem A.7

The prey species in the proposed model system (2.3) becomes extinct if $\alpha < \beta$, whereas the predator population becomes extinct if $\epsilon < \delta$.

Note:  The proof of all the theorems stated in Appendix is trivial, so we have omitted it for the sake of brevity.