A general class of predation models with multiplicative Allee effect

Abstract

A class of models of predator–prey interaction with Allee effect on the prey population is presented. Both the Allee effect and the functional response are modelled in the most simple way by means of general terms whose conveniently chosen mathematical properties agree with, and generalise, a number of concrete Leslie–Gower-type models. We show that this class of models is well posed in the sense that any realistic solution is bounded and remains non-negative. By means of topological equivalences and desingularization techniques, we find specific conditions such that there may be extinction of both species. In particular, the local basin boundaries of the origin are found explicitly, which enables one to determine the extinction or survival of species for any given initial condition near this equilibrium point. Furthermore, we give conditions such that an equilibrium point corresponding to a positive steady state may undergo saddle-node, Hopf and Bogdanov–Takens bifurcations. As a consequence, we are able to describe the dynamics governed by the bifurcated limit cycles and homoclinic orbits by means of carefully sketched bifurcation diagrams and suitable illustrations of the relevant invariant manifolds involved in the overall organisation of the phase plane. Finally, these findings are applied to concrete model vector fields; in each case, the particular relevant functions that define the conditions for the associated bifurcations are calculated explicitly.

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Acknowledgments

This work was partially funded by FONDECYT Postdoctoral Grant No. 3130497, DGIP-UTFSM Grant 12.13.10 and Proyecto Basal CMM Universidad de Chile.

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Correspondence to Pablo Aguirre.

Appendix: A normal form of the Bogdanov–Takens bifurcation

Appendix: A normal form of the Bogdanov–Takens bifurcation

Consider a planar vector field

$$\begin{aligned} \dot{\mathbf {x}}=f(\mathbf {x},\mu ), \mathbf {x}\in \mathbb {R}^2, \mu \in \mathbb {R}^2, \end{aligned}$$
(36)

where \(f\) is smooth enough. Assume that the origin \(\mathbf {x}=0\) of (36) is an equilibrium with two zero eigenvalues \(\lambda _{1,2}=0\) at \(\mu =0\), and such that the Jacobian \(D_{\mathbf {x}}f(0,0)\) is nilpotent and different from the null matrix. Equation (36) can be written at \(\mu =0\) in the form

$$\begin{aligned} \dot{\mathbf {x}}=D_{\mathbf {x}}f(0,0)\mathbf {x}+F(\mathbf {x}), \end{aligned}$$
(37)

where \(F(\mathbf {x})\) contains all the quadratic and higher order terms \(O(||\mathbf {x}||^2)\).

The matrix \(D_{\mathbf {x}}f(0,0)\) has a single linearly independent eigenvector \(\mathbf {v}_1\) that corresponds to the repeated eigenvalue 0. In addition, one can find a generalised eigenvector \(\mathbf {v}_2\) as a solution of the equation \(D_{\mathbf {x}}f(0,0)\mathbf {v}_2=\mathbf {v}_1\).

Let \(\mathbf {P}=[\mathbf {v}_1, \mathbf {v}_2]\) be the matrix whose columns are the (linearly independent) vectors \(\mathbf {v}_1\) and \(\mathbf {v}_2\). Hence, the change of coordinates

$$\begin{aligned} \mathbf {y} = \mathbf {P}^{-1}\mathbf {x} \end{aligned}$$
(38)

maps the vector field \(f\) to a \(\mathcal {C}^{\infty }\)-conjugated system defined as

$$\begin{aligned} g=\mathbf {P}^{-1}\circ f\circ \mathbf {P}. \end{aligned}$$
(39)

In particular, at \(\mu =0\), system (37) takes the form

$$\begin{aligned} \dot{\mathbf {y}}= D_{\mathbf {y}}g(0,0)\mathbf {y}+ \left( \mathbf {P}^{-1}\circ F\circ \mathbf {P}\right) (\mathbf {y}), \end{aligned}$$
(40)

where \(D_{\mathbf {y}}g(0,0)=\left( \begin{array}{cc} 0 &{} 1 \\ 0 &{} 0 \\ \end{array} \right) \).

Expanding (39) as a Taylor series with respect to \(\mathbf {y}=(y_1,y_2)\) around \((y_1,y_2)=(0,0)\), one obtains

$$\begin{aligned} \dot{y}_1&= y_2+a_{00}(\mu )+a_{10}(\mu )y_1+a_{01}(\mu )y_2\\&\quad + \frac{1}{2}a_{20}(\mu )y_1^2+a_{11}(\mu )y_1y_2\\&\quad +\frac{1}{2}a_{02}(\mu )y_2^2+O(||\mathbf {y}||^3),\\ \dot{y}_2&= b_{00}(\mu )+b_{10}(\mu )y_1+b_{01}(\mu )y_2\\&\quad + \frac{1}{2}b_{20}(\mu )y_1^2+b_{11}(\mu )y_1y_2\\&\quad +\frac{1}{2}b_{02}(\mu )y_2^2+O(||\mathbf {y}||^3), \end{aligned}$$

where the coefficients \(a_{ij}(\mu )\) and \(b_{ij}(\mu )\) are smooth functions which can be found from (36), (38) and (39). In particular, at \(\mu =0\), from (37) and (40), we have \( a_{00}(0)=a_{10}(0)=a_{01}(0)=b_{00}(0)=b_{10}(0)=b_{01}(0)=0. \)

In this setting, one can prove the following result for the normal form of the Bogdanov–Takens bifurcation:

Theorem

Suppose that the planar system (36) has, at \(\mu =0\), an equilibrium at the origin \(\mathbf {x}=0\) with a double zero eigenvalue \(\lambda _{1,2}(0)=0\). Assume that the following genericity conditions are satisfied:

  1. i.

    The Jacobian \(D_{\mathbf {x}}f(0,0)\) is not the null matrix;

  2. ii.

    \(a_{20}(0)+b_{11}(0)\ne 0;\)

  3. iii.

    \(b_{20}(0)\ne 0\);

  4. iv.

    The map

    $$\begin{aligned} (\mathbf {x},\mu )\mapsto \big (f(\mathbf {x},\mu ),\mathrm{tr}D_{\mathbf {x}}f(\mathbf {x},\mu ),\mathrm{det}D_{\mathbf {x}}f(\mathbf {x},\mu )\big ) \end{aligned}$$

    is regular at \((\mathbf {x},\mu )=(0,0)\in \mathbb {R}^4.\)

Then, there exists a smooth invertible change of parameters, such that the vector field \(f\), in a sufficiently small neighbourhood of \((\mathbf {x},\mu )=(0,0)\), is topologically equivalent to one of the following normal forms:

$$\begin{aligned} \left\{ \begin{array}{rcl} \dot{\xi }_1 &{} = &{}\xi _2, \\ \dot{\xi }_2 &{} = &{} \beta _1+\beta _2 \,\xi _2 +\xi _2^2+s\, \xi _1\xi _2,\\ \end{array} \right. \end{aligned}$$
(41)

where \(s=b_{20}(0)\big (a_{20}(0)+b_{11}(0)\big )=\pm 1\).

The construction of the normal form (41) was first developed by Bogdanov [10]. An equivalent normal form was introduced simultaneously by Takens in [42]. The interested reader can also find the proof of this theorem in [13, 33], as well as further details.

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Aguirre, P. A general class of predation models with multiplicative Allee effect. Nonlinear Dyn 78, 629–648 (2014). https://doi.org/10.1007/s11071-014-1465-3

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Keywords

  • Predator–prey model
  • Allee effect
  • Bifurcation analysis