# Analytical derivation of delayed prey-predator model with hunting-and-resting delay

Mostafa Adimy, Malay Banerjee

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Mostafa Adimy, Malay Banerjee. Analytical derivation of delayed prey-predator model with hunting-and-resting delay. Chaos, Solitons & Fractals, 2025, 195, pp.116272. 10.1016/j.chaos.2025.116272. hal-05447851

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Submitted on 8 Jan 2026

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# Analytical derivation of delayed prey-predator model with hunting-and-resting delay

Mostafa Adimy$^{a,*}$, Malay Banerjee$^b$

## Abstract

We investigate a prey-predator model based on a general Gause type system. We take for the predator two phases into account, the hunting phase and the resting one. We suppose that the predators stop hunting after they catch the prey. Then they enter the resting phase where they stay for a fixed limited time. The resulting mathematical model is a system of two age-structured partial differential equations. By integrating this system over age and using the characteristics method, we reduce it to a delay differential system, and we investigate the existence and stability of the steady states. In particular, we have shown that the introduction of the delay (the duration of the resting phase) stabilizes the coexistence equilibrium.

**Keywords:** Delay differential equation, age-structure PDE, prey-predator, stability, Hopf bifurcation

## 1. Introduction

Classical Rosenzweig-MacArthur model, [1], falls under the well-known class of Gause type prey-predator model [2]. General form of Gause type prey-predator model is given by

$$x'(t) = xf(x) - yp(x), \quad y'(t) = eyp(x) - q(y)y,$$

where $x(t)$ and $y(t)$ are the prey and predator population densities respectively at time $t$. $f(x)$ is the per capita prey growth rate in the absence of predator, $p(x)$ is prey-dependent functional response, $q(y)$ is the density dependent per capita mortality rate of the predator and $e$ is conversion efficiency ($0 < e < 1$).

In Gause type prey-predator models, the growth rate of predator is dependent upon the prey density as the consumed prey biomass supports the growth of the predator due to the birth of new off springs. This conversion is not instantaneous rather mediated by some time lag which is known as gestation delay [3, 4]. Delayed Gause type prey-predator model with maturation delay is given by

$$x'(t) = x(t)f(x(t)) - y(t)p(x(t)), \quad y'(t) = e \exp(-\delta\tau)y(t - \tau)p(x(t - \tau)) - q(y(t))y(t),$$

<footnote>$^a$INRIA, ICJ UMR 5208, CNRS, Ecole Centrale de Lyon, INSA Lyon, Université Claude Bernard Lyon 1, Université Jean Monnet, 69603 Villeurbanne, France</footnote>

<footnote>$^b$Department of Mathematics & Statistics, Indian Institute of Technology Kanpur, Kanpur - 208016, India</footnote>

<footnote>*Corresponding author</footnote>

<footnote>Email address: mostafa.adimy@inria.fr (Mostafa Adimy)</footnote>

<footnote>Preprint submitted to Elsevier</footnote>

<footnote>January 8, 2026</footnote>
here $\tau$ is maturation delay (discrete) and $\delta$ is the juvenile predator's mortality rate.

The main objective of the present work is to derive, using age-structured type PDEs, a delayed prey-predator model where the discrete time delay models the hunting-and-resting phase of the predator.

## 2. Model formulation

We suppose that the growth of the prey population $x(t)$ is given by $x'(t) = xf(x)$, where $f(x)$ is the per capita prey growth rate in the absence of predator. We will take the logistic equation

$$f(x) = r(1 - x/K). \qquad (2.1)$$

In the presence of the predator, the growth equation of $x(t)$ will be different, involving negative feedback term due to the predation of prey by their predators. We assume that the predator may be in one of the two phases: hunting and resting. We denote respectively by $u(t, a)$ and $v(t, a)$ the population densities of predators in hunting and resting phase, where $a \ge 0$ represents the time spent by a predator in one of the two phases. We assume that the duration of the resting phase is limited and is the same for all predators.

The predators stop hunting for prey after successful catch, the predation rate is described by the functional response and denoted by $\gamma(x(t))$. Here we choose the classical Rosenzweig-MacAurthur functional response

$$\gamma(x) = \alpha x/(1 + \beta x). \qquad (2.2)$$

The mortality rate of the predators during this phase is $\delta$. We can write the following equation for the predators in hunting phase

$$u_t + u_a = -\gamma(x(t))u - \delta u, \quad a > 0,$$

with the condition $\lim_{a \to +\infty} u(t, a) = 0$. The total population of predators hunting at time $t$ is given by

$$y(t) = \int_{0}^{+\infty} u(t, a) da.$$

We suppose that the duration of the resting phase, where the predator is not hunting, is $\tau \ge 0$. The mortality rate during this phase is $\mu$. It is smaller than the mortality rate during the hunting phase ($\delta \ge \mu$). For the predators in resting phase

$$v_t + v_a = -\mu v, \quad 0 < a < \tau.$$

At the end of the resting phase $a = \tau$, the predator starts immediately to hunt

$$u(t, 0) = v(t, \tau).$$

The total resting predator population at time $t$ is given by

$$z(t) = \int_{0}^{\tau} v(t, a) da.$$

It is assumed that the predators $u(t, a)$ stop hunting when they catch prey $x$ with a rate $\gamma(x)$. Then, they enter the resting phase

$$v(t, 0) = (1 + e) \int_{0}^{+\infty} \gamma(x(t))u(t, a) da = (1 + e)\gamma(x(t))y(t).$$
The parameter $0 < e < 1$ represents the conversion efficiency of the predator.

By integrating the age-structured type equation of $v(t, a)$ over the age and using the boundary condition, we get the following ordinary differential equation

$$z'(t) = -\mu z(t) + (1 + e)\gamma(x(t))y(t) - v(t, \tau).$$

From the age-structured type equation of $u(t, a)$, we obtain

$$y'(t) = -\delta y(t) - \gamma(x(t))y(t) + v(t, \tau).$$

We have to compute $v(t, \tau)$ using the method of characteristics, for $t \ge \tau$. That is

$$v(t, \tau) = e^{-\mu\tau}v(t - \tau, 0) = e^{-\mu\tau}(1 + e)\gamma(x(t - \tau))y(t - \tau).$$

In the presence of predators, the time evolution of the prey population $x(t)$ is given by

$$x'(t) = x(t)f(x(t)) - \gamma(x(t))y(t).$$

We derive the following system for the prey-predator interaction described by the nonlinear delay differential equations,

$$\begin{cases} x'(t) = x(t)f(x(t)) - \gamma(x(t))y(t), \\ y'(t) = -\delta y(t) - \gamma(x(t))y(t) + e^{-\mu\tau}(1+e)\gamma(x(t-\tau))y(t-\tau), \\ z'(t) = -\mu z(t) + (1+e)\gamma(x(t))y(t) - e^{-\mu\tau}(1+e)\gamma(x(t-\tau))y(t-\tau). \end{cases}$$

Instead of the equation of resting predators $z(t)$, we introduce the equation of the total predator population $w(t) = y(t) + z(t)$

$$w'(t) = -\mu w(t) - (\delta - \mu)y(t) + e\gamma(x(t))y(t).$$

As $x$ and $y$ are independent on $w$, we consider the system without $z$ (or $w$)

$$\begin{cases} x'(t) = x(t)f(x(t)) - \gamma(x(t))y(t), \\ y'(t) = -\delta y(t) - \gamma(x(t))y(t) + e^{-\mu\tau}(1+e)\gamma(x(t-\tau))y(t-\tau). \end{cases} \quad (2.3)$$

It is clear that solutions of initial value problem (2.3) with nonnegative initial conditions exist, are unique and are nonnegative on $[0, +\infty)$.

### 3. Steady states and their stability

Using (2.1) and (2.2), the obtain the steady states

$$x^*[r(1 - x^*/K) - \alpha y^*/(1 + \beta x^*)] = 0, \quad y^*[-\delta + ((1 + e)e^{-\mu\tau} - 1)\alpha x^*/(1 + \beta x^*)] = 0.$$

The model (2.3) admits an extinction steady-state $E_0(0, 0)$, predator free steady-state $E_1(K, 0)$ and a coexistence steady-state $\bar{E}(\bar{x}, \bar{y})$ whose components are given by

$$\bar{x}(\tau) = \delta/(\alpha((1+e)e^{-\mu\tau} - 1) - \beta\delta), \quad \bar{y}(\tau) = r(1 - \bar{x}/K)(1 + \beta\bar{x})/\alpha.$$
$\bar{E}(\bar{x}, \bar{y})$ exists if and only if

$$e > e_{\min} := \delta/\gamma(K) \quad \text{and} \quad 0 \le \tau < \tau_{\max} := \mu^{-1} \ln((1+e)/(1+e_{\min})).$$

Linearized around $(x^*, y^*)$

$$\left\{ \begin{array}{l} u'(t) = [r(1-x^*/K) - (r/K)x^* - \gamma'(x^*)y^*]u(t) - \gamma(x^*)v(t), \\ v'(t) = -\gamma'(x^*)y^*u(t) - (\delta + \gamma(x^*))v(t) + e^{-\mu\tau}(1+e)[\gamma'(x^*)y^*u(t-\tau) + \gamma(x^*)v(t-\tau)]. \end{array} \right.$$

The characteristic equation

$$\Delta(\lambda) := \det(\lambda I - A - e^{-\lambda\tau}B) = 0,$$

with

$$A = \begin{pmatrix} r(1 - 2x^*/K) - \gamma'(x^*)y^* & -\gamma(x^*) \\ -\gamma'(x^*)y^* & -(\delta + \gamma(x^*)) \end{pmatrix} \quad \text{and} \quad B = \xi_{\tau} \begin{pmatrix} 0 & 0 \\ \gamma'(x^*)y^* & \gamma(x^*) \end{pmatrix},$$

where $\xi_{\tau} := e^{-\mu\tau}(1+e)$. Then, the characteristic equation becomes

$$(\lambda - (r(1 - 2x^*/K) - \gamma'(x^*)y^*))(\lambda + \delta + \gamma(x^*) - \xi_{\tau}\gamma(x^*)e^{-\lambda\tau}) - \gamma(x^*)(\gamma'(x^*)y^* - \xi_{\tau}\gamma'(x^*)y^*e^{-\lambda\tau}) = 0.$$

### 3.1. Stability of the trivial steady states

The eigenvalues of the linearized system around the trivial steady state $E_0(0, 0)$ are $-\delta$ and $r$. Then $E_0(0, 0)$ is always a saddle point, which is an unstable equilibrium point.

The characteristic equation corresponding to the trivial steady state $E_1(K, 0)$ is

$$(\lambda + r)(\lambda + \delta + \gamma(K) - \xi_{\tau}\gamma(K)e^{-\lambda\tau}) = 0.$$

The eigenvalues are $\lambda = -r < 0$ and the roots of the equation

$$\lambda + \delta + \gamma(K) - \xi_{\tau}\gamma(K)e^{-\lambda\tau} = 0.$$

The last equation has the form $\lambda - a - be^{-\lambda\tau} = 0$, with $a = -\delta - \gamma(K) < 0$ and $b = \xi_{\tau}\gamma(K) > 0$. We always have $a - b < 0$. On the other hand,

$$a + b = -\delta - (1 - \xi_{\tau})\gamma(K) = -\delta - (1 - e^{-\mu\tau}(1 + e))\gamma(K).$$

Then, $a + b < 0$ if and only if $\tau > \tau_{\max}$. This means that the characteristic roots (see [5]) have negative real part if and only if $\tau > \tau_{\max}$. In particular, if $e < e_{\min}$ we have always $\tau > \tau_{\max}$. We conclude that if $\tau > \tau_{\max}$, the predator goes extinct.

### 3.2. Stability analysis of the coexistence steady state

The coexistence steady state $\bar{E}(\bar{x}, \bar{y})$ is given by

$$\bar{x} = \delta/(\alpha((1+e)e^{-\mu\tau} - 1) - \beta\delta), \quad \bar{y} = r(1 - \bar{x}/K)(1 + \beta\bar{x})/\alpha.$$

under the condition

$$e > e_{\min} := \delta/\gamma(K) \quad \text{and} \quad 0 \le \tau < \tau_{\max} := \mu^{-1} \ln((1+e)/(1+e_{\min})).$$
First, we study the stability of the coexistence steady state $\bar{E}(\bar{x}, \bar{y})$ in the case $\tau = 0$, under the condition $e > e_{\min}$. The system (2.3) becomes

$$x'(t) = x(t)f(x(t)) - \gamma(x(t))y(t), \quad y'(t) = -\delta y(t) + e\gamma(x(t))y(t), \quad (3.4)$$

with the coexistence steady state

$$\bar{x} = \delta/(\alpha e - \beta \delta), \quad \bar{y} = r(1 - \bar{x}/K)(1 + \beta \bar{x})/\alpha.$$

Suppose that $\beta K > 1$ and set $e_H := \beta \delta(\beta K + 1)/\alpha(\beta K - 1)$. Then, the coexistence steady state $(\bar{x}, \bar{y})$ of (3.4) is a sink (stable) if

$$e_{\min} < e < e_H$$

and the system (3.4) undergoes a super-critical Hopf-bifurcation at $e = e_H$. If $\beta K \le 1$, then $(\bar{x}, \bar{y})$ is stable for all $e > e_{\min}$. What is the impact of the delay on the stability of the coexistence steady state $\bar{E}(\bar{x}, \bar{y})$, for $e > e_{\min}$ and $0 \le \tau < \tau_{\max}$?

First, it is clear that $\tau \mapsto \bar{x}(\tau)$ is increasing and $\tau \mapsto \bar{y}(\tau)$ is decreasing with $\lim_{\tau \to \tau_{\max}} \bar{x}(\tau) = K$ and $\lim_{\tau \to \tau_{\max}} \bar{y}(\tau) = 0$. The characteristic equation at $\bar{E}(\bar{x}, \bar{y})$ is a second order transcendental equation of the form

$$\Delta(\lambda) = P_{\tau}(\lambda) + Q_{\tau}(\lambda)e^{-\lambda\tau} = 0, \quad (3.5)$$

where $P_{\tau}$ and $Q_{\tau}$ are polynomials of second and first order, in which the coefficients depend on the delay $\tau$. A standard condition for the occurrence of a limit cycle in the system (2.3) is that a pair of complex conjugate roots of (3.5) intersects the imaginary axis of the complex plane (with a nonzero imaginary part) when the delay $\tau$ is varied. Determining if this happens and for what value of the delay is difficult because the coefficients of (3.5) depend on the delay. We refer the reader to [6] for more details on the conditions for the existence of such a stability change. We will check the effect of the delay $\tau \in [0, \tau_{\max})$ by numerically simulating the system.

We consider a fixed parameter set, $r = 1$, $\beta = 0.5$, $\delta = 0.2$, $K = 10$ and $\alpha = 0.3$. We can calculate $e_{\min} = 0.4$ and $e_H = 0.5$. The system (3.4), $\tau = 0$, undergoes a super-critical Hopf-bifurcation at $e = e_H = 0.5$. Our main aim is to show the stabilizing effect of the delay $\tau$ on the system dynamics. For this purpose we choose two different values of $e$: $e = 0.45 \equiv e_1$ and $e = 0.6 \equiv e_2$ such that $e_{\min} < e_1 < e_H < e_2$. For $e = e_1$, $\bar{E}(5.71, 5.51)$ is stable for the model (3.4), whereas for $e = e_2$, $\bar{E}(2.5, 5.63)$ is unstable and is surrounded by a stable limit cycle. Now we consider the delayed model (2.3) and fix $\mu = 0.2$. One can calculate $\bar{E}(\bar{x}(\tau), \bar{y}(\tau))$ is feasible whenever $\tau \le \tau_{\max}(e_1) = 0.1755$ and $\tau \le \tau_{\max}(e_2) = 0.6677$ for $e = e_1$ and $e = e_2$ respectively. Numerical simulation reveals that the coexistence equilibrium point $\bar{E}(\bar{x}(\tau), \bar{y}(\tau))$ remain stable for all $0 < \tau < \tau_{\max}(e_1)$ (see Fig. 1(a)). On the other hand, when $e = e_2$, the coexistence equilibrium point $\bar{E}(\bar{x}(\tau), \bar{y}(\tau))$ is unstable and surrounded by a stable limit cycle for $0 < \tau < 0.36 \equiv \tau_H$ and is stable for $0.36 < \tau < \tau_{\max}(e_2) = 0.6677$ (see Fig. 1(b)). $\bar{E}(\bar{x}(\tau), \bar{y}(\tau))$ coincides with the axial equilibrium point $E_1(10, 0)$ at $\tau = \tau_{\max}$ in both the cases.

## 4. Discussion

The incorporation of discrete-time delay in the interacting population model is well known and often leads to destabilization of coexistence steady-state. Sometimes one finds switching of stability due to the variation of the delay parameter; however, there is a lack of biological evidence toward stability switching. This work derived a delay differential equation model for prey-predator interaction with specialist predators. The discrete-time delay represents the
<figure><img src="image_3.png" /><figcaption>Figure 1: (a) Components of stable coexistence equilibrium point are plotted against τ as bifurcation parameter for $e = e_1 = 0.45$. (b) Unstable (broken curve) and stable components of coexistence equilibrium point are shown with varying τ for $e = e_2 = 0.6$. The maxima and minima of stable limit cycles are also plotted when τ varies in $[0, τ_H]$ along with unstable components. Other parameter values are mentioned in the text. Blue and black curves correspond to x and y components respectively.</figcaption></figure>

hunting-and-resting time for the predator, and the delay model is derived mechanistically from an age-structured PDEs. Linear stability analysis indicates that the discrete-time delay does not affect the stability of boundary equilibrium points. Further, the stable coexistence equilibrium point for the non-delay model remains stable when we switch to the delayed model. On the other hand, the hunting-and-resting time delay can drive the oscillatory coexistence scenario to stable steady-state coexistence. Our future goal will be to consider the effect of the proposed delay on the dynamics of prey-predator models with other types of functional responses and in the presence of generalist predators.

## Acknowledgement

A part of this work was carried out when the second author have visited INRIA, Team Drac ula, Lyon France under the financial support of Associate Team (INRIA program) EQA-071905.

## References

[1] M. Rosenzweig and R. MacArthur, Graphical representation and stability conditions of predator-prey interaction, American Naturalist (1963) 97, 209–223.

[2] H.I. Freedman, Deterministic Mathematical Models in Population Ecology, Marcel Dekker, New York, 1980.

[3] M. Banerjee and Y. Takeuchi, Maturation delay for the predators can enhance stable coexistence for a class of prey-predator models. Journal of Theoretical Biology, 412, 154–171 (2015).

[4] S.A. Gourley and Y. Kuang, A stage structured predator-prey models and its dependence of maturation delay and death rate. Journal of Mathematical Biology, 46, 188–200 (2004).

[5] J.K. Hale and S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, Springer-Verlag New York, 1993.

[6] G. Fan, M. Min-Oo and G.S.K. Wolkowicz, Hopf bifurcation of delay Differential equations with delay dependent parameters. Canadian Applied Mathematics Quarterly, 17(1), 37–60 (2009).