Abstract
The extinction of species is a major threat to the biodiversity. The species exhibiting a strong Allee effect are vulnerable to extinction due to predation. The refuge used by species having a strong Allee effect may affect their predation and hence extinction risk. A mathematical study of such behavioral phenomenon may aid in management of many endangered species. However, a little attention has been paid in this direction. In this paper, we have studied the impact of a constant prey refuge on the dynamics of a ratio-dependent predator–prey system with strong Allee effect in prey growth. The stability analysis of the model has been carried out, and a comprehensive bifurcation analysis is presented. It is found that if prey refuge is less than the Allee threshold, the incorporation of prey refuge increases the threshold values of the predation rate and conversion efficiency at which unconditional extinction occurs. Moreover, if the prey refuge is greater than the Allee threshold, situation of unconditional extinction may not occur. It is found that at a critical value of prey refuge, which is greater than the Allee threshold but less than the carrying capacity of prey population, system undergoes cusp bifurcation and the rich spectrum of dynamics exhibited by the system disappears if the prey refuge is increased further.
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Appendices
Appendix
Positivity and Boundedness of the Model Solutions
From the model system (1), we have
and
From the above, we can see that \(x(t) \ge 0\) and \(y(t) \ge 0\) whenever \(x(0) > 0\) and \(y(0) >0\). Thus, all the solutions starting in the interior of the positive quadrant remain in it for all \(t \ge 0\).
Now, we show the boundedness of the model solutions. For this, consider Eq. (25), which can be written as
where,
Now, two cases arise:
Case I: \(x(0)\in (0,K)\)
In this case, we claim that \(x(t) \le K\) for all \(t \ge 0\). Otherwise, there exists two positive real numbers \(t_1\) and \(t_2\) such that \(x(t_1)=K\) and \(x(t)> K\) for all \(t \in (t_1, t_2)\). Then, for all \(t \in (t_1, t_2)\),
as \(F (x(t), y(t))<0\) for all \(t \in (t_1, t_2)\). This contradicts our hypothesis. Hence \(x(t) \le K\) for all future time.
Case II: When \(x(0)>K\).
In this case, as long as \(x(t)\ge K\)
as \(F (x(t), y(t))\le 0\) for \(x(t)\ge K\).
Combining both the above cases, it can be concluded that for any positive solution
Now, from model system (1), we have
Let \(\xi = max_{t\ge 0}r x \left( 1- \frac{x}{K}\right) (x-\theta ) + d x\), then
Using Gronwall’s inequality, we have
For large enough t, we can write
for arbitrary \(\epsilon >0\). Since x(t) is bounded, (30) implies that y(t) is also bounded.
Stability of Bifurcating Periodic Solutions
Applying the transformation
the model system (3) reduces to the following system
where
The system (32) can be written as
where,
The eigenvector v of Jacobian matrix \(J_{E_2^*}\) corresponding to the eigenvalue \(i \omega _0\) at \(a=a_c\) is found to be \( v=( a_{01}, i \omega _0-a_{10})^T\). Now define
Let \(Z=A Y\) or \(Y=A^{-1} Z\), where \(Y= (y_1, y_2)^T\). Under this linear transformation, system (33) becomes
This can be written as,
where
In order to determine the stability and direction of the periodic solution, we need to calculate the following quantity called Lyapunov coefficient (Wiggins 1990).
where \(\displaystyle F^k_{ij}=\left[ \frac{\partial F^k}{\partial y_i \partial y_j}\right] _{(0,0;a_c)}\) and \(\displaystyle F^k_{ijl}=\left[ \frac{\partial F^k}{\partial y_i \partial y_j \partial y_l}\right] _{(0,0; a_c)}\).
The bifurcating periodic solutions are orbitally stable or unstable according as \(l_1<0\) or \(l_1>0\).
Conditions for the Non-degeneracy of BT Bifurcation
In the following, we will show that the conditions for non-degeneracy of BT bifurcation are satisfied using the algorithm given in (Kuznetsov 1998). Consider the system
where \(\nu _1\) and \(\nu _2\) are small. When \(\nu _1=0\) and \(\nu _2=0\), system (36) has one positive equilibrium \((x^*, y^*)\), which is a cusp of codimension 2.
Let \(u_1 = x - x^*\) and \(u_2 = y - y^*\). Then, system (36) becomes
where
Now, using the affine transformation \(w_1=u_1\) and \(w_2=c_{10}u_1+c_{01} u_2\), the system (37) reduces to
where
The degeneracy conditions of BT bifurcation are
We find that
and
Thus, the degeneracy conditions of BT bifurcation are satisfied. The bifurcation structure of the BT point is given by the following quantity
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Verma, M., Misra, A.K. Modeling the Effect of Prey Refuge on a Ratio-Dependent Predator–Prey System with the Allee Effect. Bull Math Biol 80, 626–656 (2018). https://doi.org/10.1007/s11538-018-0394-6
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DOI: https://doi.org/10.1007/s11538-018-0394-6