Complex dynamics in an eco-epidemiological model with the cost of anti-predator behaviors

Abstract

In this paper, we investigate the complex dynamics of a predator–prey model with disease in the prey, which is characterized by the reduction of prey growth rate due to the anti-predator behavior. The value of this study lies in two aspects: Mathematically, it provides the existence and the stability of the equilibria and gives the existence of Hopf bifurcation. And epidemiologically, we find that the influence of the fear factor is complex: (i) The level of the population density decreases with the increasing of the fear factor; (ii) the cost of fear can destabilize the stability and benefit the emergency of the periodic behavior; and (iii) the high level of fear can induce the extinction of the predator. These results may enrich the dynamics of the eco-epidemiological systems.

Appendices

Appendix A: The sign of $\xi (\mathscr {R}_0)$

Proof

Recall

$$\begin{aligned} \xi (\mathscr {R}_0):=b\delta \mu \mathscr {R}_{0}^2-r(r cp-b\mu )\mathscr {R}_{0}+r^2cp, \end{aligned}$$

so we can discuss the sign of $\xi (\mathscr {R}_0)$ in the following two cases.

Case 1 $rcp\le b\mu$. In this case, $\xi (\mathscr {R}_0)>0.$

Case 2 $rcp>b\mu$.

Set

$$\begin{aligned} \varDelta (b):= & {} (rcp-b\mu )^2-4cpb\delta \mu \\= & {} \mu ^2b^2-2cp\mu (r+2\delta )b+r^2c^2p^2. \end{aligned}$$

Note that $(-2cp\mu (r+2\delta ))^2-4\mu ^2r^2c^2p^2=16c^2\delta \mu ^2p^2(r+\delta )>0$, then $\varDelta (b)$ has two positive roots $b_-$ and $b_+$, which are defined as in (7).

Case 2-1 If $b_-<b<b_+$ holds, we have $\varDelta (b)<0,$ which implies that $\xi (\mathscr {R}_0)>0.$

Case 2-2 If $b=b_\pm$ holds, we have $\varDelta (b)=0.$ Thus, when $\mathscr {R}_0=\dfrac{2rcp}{rcp-b\mu }$, we have $\xi (\mathscr {R}_0)=0$; when $\mathscr {R}_0\ne \dfrac{2rcp}{rcp-b\mu }$, we have $\xi (\mathscr {R}_0)>0$.

Case 2-3 If $0<b<b_-$ or $b>b_+$ holds, we have $\varDelta (b)>0.$

Then, one can see that $\xi (\mathscr {R}_0)$ has two positive roots $\mathscr {R}_0^-$ and $\mathscr {R}_0^+$, which are defined as in (8). In fact,

$$\begin{aligned} 1<\dfrac{2rcp}{rcp-b\mu +\sqrt{(rcp-b\mu )^2-4cpb\delta \mu }}=\mathscr {R}_0^-. \end{aligned}$$

Hence, if $\mathscr {R}_0=\mathscr {R}_0^\pm$ holds, we have $\xi (\mathscr {R}_0)=0;$ if $1<\mathscr {R}_0^-<\mathscr {R}_0<\mathscr {R}_0^+$ holds, we have $\xi (\mathscr {R}_0)<0;$ and if $1<\mathscr {R}_0<\mathscr {R}_0^-$ or $\mathscr {R}_0>\mathscr {R}_0^+$ holds, we have $\xi (\mathscr {R}_0)>0.$

Therefore, we can determine the sign of $\xi (\mathscr {R}_0)$ in three cases as follows:

(1) when $rcp>b\mu$, if $0<b<b_-$ or $b>b_+$, and $1<\mathscr {R}_0^-<\mathscr {R}_0<\mathscr {R}_0^+$ hold, we have $\xi (\mathscr {R}_0)<0;$

(2) when $rcp>b\mu$, if $0<b\le b_-$ or $b\ge b_+$, and $\mathscr {R}_0=\mathscr {R}_0^\pm$, we have $\xi (\mathscr {R}_0)=0.$

(3) if one of the following inequalities holds:

   (3-1) $rcp\le b\mu$;

   (3-2) $rcp> b\mu$,

      (i) $b_-<b<b_+$;

      (ii) $b=b_\pm$ and $\mathscr {R}_0\ne \dfrac{2rcp}{rcp-b\mu }$;

      (iii) $0<b<b_-$ or $b>b_+$, and $1<\mathscr {R}_0<\mathscr {R}_0^-$ or $\mathscr {R}_0>\mathscr {R}_0^+$,

we have $\xi (\mathscr {R}_0)>0.$ $\square$

Appendix B: The Proof of Theorem 3

Proof

The Jacobin matrix of model (3) around $E_{2}=\left( \frac{r}{b\mathscr {R}_0}, \frac{r^2(\mathscr {R}_0-1)}{b\mathscr {R}_0(\delta \mathscr {R}_0+r)}, 0\right)$ is given as

$$\begin{aligned} J_2=\left( \begin{array}{ccc} -\frac{r}{\mathscr {R}_0}&{} -\frac{r}{\mathscr {R}_0}-\delta &{} -\frac{akr}{b\mathscr {R}_0}\\ \frac{\delta r(\mathscr {R}_0-1)}{\delta \mathscr {R}_0+r} &{} 0 &{}-\frac{pr^2(\mathscr {R}_0-1)}{b\mathscr {R}_0(\delta \mathscr {R}_0+r)}\\ 0 &{} 0&{} -\frac{\xi (\mathscr {R}_0)}{b\mathscr {R}_0(\delta \mathscr {R}_0+r)} \end{array} \right) . \end{aligned}$$

Hence, the characteristic equation of $J_2$ is given as

$$\begin{aligned}&\Bigl (\lambda ^2\mathscr {R}_0+r\lambda +r\delta (\mathscr {R}_0-1)\Bigr ) (-b\mathscr {R}_0(\delta \mathscr {R}_0+r)\lambda \\&\quad -b\delta \mu \mathscr {R}_{{0}}^2 +r(rcp-b\mu )\mathscr {R}_{{0}}-r^2cp)=0. \end{aligned}$$

Set

$$\begin{aligned} \phi (\lambda ):= & {} -b\mathscr {R}_0(\delta \mathscr {R}_0+r) \lambda -b\delta \mu \mathscr {R}_{{0}}^2\\&\quad +r(rcp-b\mu )\mathscr {R}_{{0}}-r^2cp\\&=-b\mathscr {R}_0(\delta \mathscr {R}_0+r)\lambda -\xi (\mathscr {R}_0), \end{aligned}$$

then $\phi (\lambda )$ has a unique root $-\frac{\xi (\mathscr {R}_0)}{b\mathscr {R}_0(\delta \mathscr {R}_0+r)}.$

Case 1 When $\xi (\mathscr {R}_0)>0$, $E_2$ is stable;

Case 2 When $\xi (\mathscr {R}_0)<0$, $E_2$ is unstable;

Case 3 When $\xi (\mathscr {R}_0)=0$, the root of $\phi (\lambda )$ is zero, which implies that $E_2$ is a singular point with higher order. In this case, we have

$$\begin{aligned} \begin{array}{ll} cpv_2-\mu &{}=\dfrac{cpr^2(\mathscr {R}_0-1)}{b\mathscr {R}_0(\delta \mathscr {R}_0+r)}-\mu \\ &{}=\dfrac{b\delta \mu \mathscr {R}_{0}^2-r(r cp-b\mu )\mathscr {R}_{0}+r^2cp}{b\mathscr {R}_0(\delta \mathscr {R}_0+r)}\\ &{}=\dfrac{\xi (\mathscr {R}_0)}{b\mathscr {R}_0(\delta \mathscr {R}_0+r)}\\ &{}=0, \end{array} \end{aligned}$$

which implies that

$$\begin{aligned} v_2=\dfrac{r^2(\mathscr {R}_0-1)}{b\mathscr {R}_0(\delta \mathscr {R}_0+r)}=\dfrac{\mu }{cp}. \end{aligned}$$

Let $u=\bar{u}-\dfrac{r}{b\mathscr {R}_0}$, $v=\bar{v}-\dfrac{\mu }{cp}$ and $w=\bar{w}$, then the model (3) becomes

$$\begin{aligned} \left\{ \begin{array}{ll} \dfrac{\mathrm {d}u}{\mathrm {d}t}&{}=-\dfrac{(bu\mathscr {R}_0+r) (b\delta \mathscr {R}_0v+dkrw+kr^2w+bru+brv)}{rb\mathscr {R}_0},\\ \dfrac{\mathrm {d}v}{\mathrm {d}t}&{}=\dfrac{(cpv+\mu )(b\delta \mathscr {R}_0u-prw)}{cpr},\\ \dfrac{\mathrm {d}w}{\mathrm {d}t}&{}=cpvw, \end{array}\right. \end{aligned}$$

(24)

where we substitute u, v, w for $\bar{u}, \bar{v}, \bar{w}$. Hence, the planar equilibrium $E_2$ moves to (0, 0, 0). The Jacobian matrix at (0, 0, 0) of (24) is

$$\begin{aligned} J_2=\left( \begin{array}{ccc} -\frac{r}{\mathscr {R}_0} &{} -\frac{b\delta \mathscr {R}_0+br}{b\mathscr {R}_0} &{} -\frac{dkr+kr^2}{b\mathscr {R}_0}\\ \frac{\mu \mathscr {R}_0b\delta }{cpr} &{} 0 &{}-\frac{\mu }{c}\\ 0 &{} 0 &{} 0 \end{array}\right) . \end{aligned}$$

(25)

Thus, the center manifold is a curve tangent to the $w-$axis. In order to obtain the approximative expression of the center manifold, we set

$$\begin{aligned} \begin{array}{ll} u=\bar{n}_1w+\bar{n}_2w^2+O(w^2),\\ v=\tilde{n}_1w+\tilde{n}_2w^2+O(w^2). \end{array} \end{aligned}$$

(26)

Then, we have

$$\begin{aligned} \begin{array}{ll} \dfrac{\mathrm {d}u}{\mathrm {d}t}=\bar{n}_1\dfrac{\mathrm {d}w}{\mathrm {d}t}+2\bar{n}_2w\dfrac{\mathrm {d}w}{\mathrm {d}t}+O(w),\\ \dfrac{\mathrm {d}v}{\mathrm {d}t}=\tilde{n}_1\dfrac{\mathrm {d}w}{\mathrm {d}t}+2\tilde{n}_2w\dfrac{\mathrm {d}w}{\mathrm {d}t}+O(w) \end{array} \end{aligned}$$

(27)

Substituting (24) and (26) into (27), we can obtain

$$\begin{aligned}&-r(b\delta \mathscr {R}_0\tilde{n}_1+br(\bar{n}_1+\tilde{n}_1)+kr(d+r))w\nonumber \\&\quad -b(b\delta \mathscr {R}_0^2\bar{n}_1\tilde{n}_1+cpr\mathscr {R}_0 \bar{n}_1\tilde{n}_1\nonumber \\&\quad +br\mathscr {R}_0\bar{n}_1^2\nonumber \\&\quad +br\mathscr {R}_0\bar{n}_1\tilde{n}_1+dkr\mathscr {R}_0\bar{n}_1\nonumber \\&\quad +kr^2\mathscr {R}_0\bar{n}_1+\delta r\mathscr {R}_0\tilde{n}_2\nonumber \\&\quad +r^2\bar{n}_2+r^2\tilde{n}_2)w^2+O(w^2)=0,\nonumber \\&\qquad \qquad (b\delta \mu \mathscr {R}_0\bar{n}_1-\mu pr)w\nonumber \\&\quad +(bc\delta p\mathscr {R}_0\bar{n}_1 \tilde{n}_1-c^2p^2r\tilde{n}_1^2\nonumber \\&\quad +b\delta \mu \mathscr {R}_0\bar{n}_2-cp^2r\tilde{n}_1)w^2+O(w^2)=0. \end{aligned}$$

(28)

Comparing the coefficients of w and $w^2$ in (28), we find that

$$\begin{aligned} \bar{n}_1= & {} \dfrac{rp}{\mathscr {R}_0b\delta },\;\;\\ \bar{n}_2= & {} \dfrac{c^2p^2r^3(d\delta k\mathscr {R}_0+\delta kr\mathscr {R}_0+pr)^2}{\mathscr {R}_0^3b^3\delta ^3(\delta \mathscr {R}_0+r)^2\mu },\;\;\\ \tilde{n}_1= & {} -\dfrac{(\delta k\mathscr {R}_0(d+r)+pr)r}{\mathscr {R}_0b\delta (\delta \mathscr {R}_0+r)},\\ \tilde{n}_2= & {} \dfrac{r^2p^2c(\delta k\mathscr {R}_0(d+r)+pr) (b\delta ^2\mu \mathscr {R}_0^3-cd\delta kr^2\mathscr {R}_0-c\delta kr^3\mathscr {R}_0 +b\delta \mu r\mathscr {R}_0^2-cpr^3)}{b^3\mathscr {R}_0^3\delta ^3(\delta \mathscr {R}_0+r)^3\mu }. \end{aligned}$$

Therefore, substituting (26) into (24), we have

$$\begin{aligned} \dfrac{\mathrm {d}w}{\mathrm {d}t}=&-cprb^2\mathscr {R}_0^2\delta ^2\mu (\delta k\mathscr {R}_0(d+r)+pr)\nonumber \\&(\delta \mathscr {R}_0+r)^2w^2+O(w^3), \end{aligned}$$

(29)

which yield that the origin (0, 0, 0) of system (24) is locally asymptotically stable. Thus, $E_2$ is locally asymptotically stable when $\xi (\mathscr {R}_0)=0$.

The proof is completed. $\square$