Uniform persistence in a prey–predator model with a diseased predator

Abstract

Following the well-extablished mathematical approach to persistence and its developments contained in Rebelo et al. (Discrete Contin Dyn Syst Ser B 19(4):1155–1170. https://doi.org/10.3934/dcdsb.2014.19.1155, 2014) we give a rigorous theoretical explanation to the numerical results obtained in Bate and Hilker (J Theoret Biol 316:1–8. https://doi.org/10.3934/dcdsb.2014.19.1155, 2013) on a prey–predator Rosenzweig–MacArthur model with functional response of Holling type II, resulting in a cyclic system that is locally unstable, equipped with an infectious disease in the predator population. The proof relies on some repelling conditions that can be applied in an iterative way on a suitable decomposition of the boundary. A full stability analysis is developed, showing how the “invasion condition” for the disease is derived. Some in-depth conclusions and possible further investigations are discussed.

Appendices

Appendix: Disease-free model

The underlying prey–predator system is a standard Rosenzweig–MacArthur model which has already been extensively studied: we refer to Rosenzweig and MacArthur (1963) for the original paper and to Hsu et al. (1978) for the stability analysis. In this Appendix we recall and discuss some known results for sake of clarity in the main dissertation and to show how condition (3) is obtained. The model reads

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} &{}{\dot{N}}=rN(1-N)-\dfrac{NS}{h+N}=N\left[ r(1-N)-\dfrac{S}{h+N}\right] \\ &{}{\dot{S}}=\dfrac{NS}{h+N}-mS=S\left[ \dfrac{N}{h+N}-m\right] \end{array}\right. } \end{aligned} \end{aligned}$$

(7)

where the Kolmogorov-like structure has been highlighted.

We identify three equilibrium points: the origin (0, 0), the trivial prey-only equilibrium (1, 0) and a non trivial one $(N^*,S^*)$ which is given by

$$\begin{aligned} N^*=\dfrac{mh}{1-m}\qquad S^*=r(1- N^*)(h+ N^*). \end{aligned}$$

The Jacobian of system (7) evaluated in the equilibrium points reads

$$\begin{aligned} {\mathbf {J}}(0,0)=\left( \begin{array}{cc}r&{}\quad 0\\ 0 &{}\quad -m\end{array}\right) ,\quad {\mathbf {J}}(1,0)=\left( \begin{array}{cc}-r&{}\quad -\dfrac{1}{h+1} \\ 0 &{}\quad \dfrac{1}{h+1}-m\end{array}\right) \end{aligned}$$

and

$$\begin{aligned} {\mathbf {J}}( N^*, S^*)=\left( \begin{array}{cc}rm\left( 1-\dfrac{1+m}{1-m}h\right) &{}\quad -m \\ r(1-m(1+h))&{}\quad 0\end{array}\right) . \end{aligned}$$

The origin is always a saddle. To avoid the stability of the trivial equilibrium it must be

$$\begin{aligned} m<\frac{1}{1+h} \end{aligned}$$

which also ensures

$$\begin{aligned} \text {det}\,{\mathbf {J}}( N^*, S^*)=mr(1-m(1+h))>0 \end{aligned}$$

so that the nontrivial equilibrium is unstable if

$$\begin{aligned} \text {tr}\,{\mathbf {J}}(N^*,S^*)=rm\left( 1-\dfrac{1+m}{1-m}h \right) >0\quad \Longleftrightarrow \quad h<\frac{1-m}{1+m} \end{aligned}$$

which is equivalent to (3). Three cases are hence given:

$\circ$:

$m>1/(1+h)$: the logistic equilibrium is stable

$\circ$:

$(1-h)/(1+h)<m<1/(1+h)$: (1, 0) is unstable and $(N^*,S^*)$ is stable

$\circ$:

$m<(1-h)/(1+h)$: both equilibrium points are unstable.

Once the last hypothesis (condition (3)) is chosen, the literature ensures that a stable and unique limit cycle bifurcates from the unstable equilibrium $(N^*,S^*)\,$, as reported in Bate and Hilker (2013). We provide a sketch the proof. First of all note that system (7) is dissipative once defined on a set

$$\begin{aligned} X=\{(x,y)\in {\mathbb {R}}_+^2:\ x+y\le k\} \end{aligned}$$

with k large enough. We could prove that the flow $\pi$ associated with (7) is uniformly persistent by means of Theorem 2 but this comes straightforwardly from (Garrione and Rebelo 2016, Theorem 3.2(b)).

The existence of the limit cycle comes from the Poincaré-Bendixson annular region Theorem, for which persistence on X is needed, while its uniqueness is proven in Cheng (1981): the stability is a consequence of the uniqueness as shown in Hsu et al. (1978) and in the more classical (Lefschetz 1963). For a general result on the limit cycles of Kolmogorov systems like (7) see (Yuan et al. 2012).

Appendix: Stability analysis

The four equilibria on the boundary pointed out in Sect. 3 are the origin, the trivial logistic equilibrium (1, 0, 0), the disease-free equilibrium $( N^*, S^*,0)$ and the limit cycle in the disease-free plane that we named $\gamma ^*$. We choose m as in (3) such that on that plane all critical points are unstable and the limit cycle is stable. We now want to study the stable manifolds of the equilibrium points.

Let us write the linearised matrix for the system (1):

$$\begin{aligned} {\mathbf {J}}(x,y,z)=\left( \begin{array}{ccc}r(1-2x)- \dfrac{h(y+z)}{(h+x)^2}&{}\quad -\dfrac{x}{h+x}&{}\quad -\dfrac{x}{h+x}\\ \dfrac{h(y+z)}{(h+x)^2}&{}\quad \dfrac{x}{h+x}-m-\beta z&{}\quad \dfrac{x}{h+x}-\beta y\\ 0&{}\quad \beta z&{}\quad \beta y-(m+\mu )\end{array}\right) . \end{aligned}$$

Evaluating in the equilibrium points:

$$\begin{aligned} {\mathbf {J}}(0,0,0)= & {} \left( \begin{array}{ccc}r&{}\quad 0&{}\quad 0\\ 0&{}\quad -m&{}\quad 0\\ 0&{}\quad 0&{}\quad -(m+\mu ) \end{array}\right) \\ {\mathbf {J}}(1,0,0)= & {} \left( \begin{array}{ccc}-r&{}\quad - \dfrac{1}{h+1}&{}\quad -\dfrac{1}{h+1}\\ 0&{}\quad \dfrac{1}{h+1}-m&{}\quad \dfrac{1}{h+1} \\ 0&{}\quad 0&{}\quad -(m+\mu )\end{array}\right) \\ {\mathbf {J}}(N^*,S^*,0)= & {} \left( \begin{array}{ccc}rm\left( 1- \dfrac{1+m}{1-m}h\right) &{}\quad -m&{}\quad -m\\ r(1-m(1+h))&{}\quad 0&{}\quad m-\beta S^*\\ 0&{}\quad 0&{}\quad \beta S^*-(m+\mu )\end{array}\right) . \end{aligned}$$

It is easy to see that

$$\begin{aligned} W^s(\{(0,0,0\})\cap X_1=\partial _x X_1=\{(x,y,z)\in X_1:\ x=0\}. \end{aligned}$$

As for (1, 0, 0), the tangent plane to its stable manifold is given by

$$\begin{aligned} k_1(1,0,0)+k_2\left( \dfrac{\mu }{(1+\mu (h+1))(m+\mu -r)},-\dfrac{1}{ 1+\mu (h+1)},1\right) ,\quad k_1,k_2\in {\mathbb {R}} \end{aligned}$$

which lies strictly outside the positive orthant except for the points in $\{(x,y,z)\in X_1:\ y=z=0\}$, all belonging to the stable manifold of (1, 0, 0) except for the origin. It holds $W^s(\{(1,0,0)\})\cap \text {int}\,X_1=\emptyset$ because an orbit belonging to $W^s(\{(1,0,0)\})$ should approach (1, 0, 0) tangentially to the above plane but this is impossible from the inside of $X_1$ as its boundary is either forward invariant ($\partial _z X_1, \partial _x X_1$) or repulsive ($\partial _y X_1$). Thus

$$\begin{aligned} W^s(\{(1,0,0)\})\cap X_1=\{(x,y,z)\in X_1:\ x>0,\ y=z=0\}. \end{aligned}$$

The first two eigenvalues $\lambda _1,\lambda _2$ for the non trivial equilibrium $(N^*,S^*,0)$ are strictly positive, as illustrated in Appendix A. $\lambda _3=\beta S^*-(m+\mu )$ could in principle possess a one dimensional stable manifold if

$$\begin{aligned} R_0^*:=\dfrac{\beta S^*}{m+\mu }<1. \end{aligned}$$

The tangent line in $(N^*,S^*,0)$ to this manifold is

$$\begin{aligned} (N^*,S^*,0)+k\left( \dfrac{\mu m}{f(\lambda _1,\lambda _2,\lambda _3)}, \dfrac{\mu (\lambda _1+\lambda _2-\lambda _3)}{f(\lambda _1,\lambda _2, \lambda _3)},1\right) ,\quad k\in {\mathbb {R}} \end{aligned}$$

where $f(\lambda _1,\lambda _2,\lambda _3)=(\lambda _1+\lambda _2)\lambda _3-(\lambda _1\lambda _2+\lambda _3^2)$. The line cuts through the disease-free face, hence the stable manifold passes through the interior of $X_1$ and can potentially lead to the extinction of the disease. To avoid this chance we ask for

$$\begin{aligned} R_0^*>1. \end{aligned}$$