Two Different Types Predator with Two Preys

M. Krishika; S. Vijaya

doi:10.22342/jims.v32i2.2229

From Equations to Action: Mathematical Models Driving Epidemics Control Strategies

M. Krishika ⁽¹⁾ , S. Vijaya ⁽²⁾

(1) Department of Mathematics, Annamalai University, India,

(2) Department of Mathematics, Annamalai University, India

https://doi.org/10.22342/jims.v32i2.2229

Issue
Vol. 32 No. 2 (2026): JUNE

Submitted
September 4, 2025

Accepted
February 8, 2026

Published
June 30, 2026

Keywords:

Generalist, specialist, Lotka-Volterra, refuge, harvesting

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Abstract

This study proposes a mathematical model involving two prey species and two predator types, where only the first prey species benefits from a refuge. The specialist predator is subjected to non-constant harvesting, reflecting fluctuating exploitation pressure. Species interactions follow a Holling type II functional response. The analysis shows that the model is bounded and identifies the equilibrium points along with their stability. Numerical results are illustrated.

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References

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S. B. Hsu and S. P. Hubbell, “Two predators competing for two prey species: An analysis of macarthur’s model,” Mathematical BioSciences, 2025. https://hal.science/hal-05011782v2.

R. Bhattacharyya and B. Mukhopadhyay, “Spatial dynamics of nonlinear prey-predator models with prey migration and predator switching,” Ecological Complexity, vol. 3, no. 2, pp. 160–169, 2006. https://doi.org/10.1016/j.ecocom.2006.01.001.

V. Hadziabdic, M. Mehulji, and J. Bektesevic, “Lotka-volterra model with two predators and their prey,” TEM Journal, vol. 6, no. 1, pp. 132–136, 2017. http://dx.doi.org/10.18421/TEM61-19.

Z. Ma, W. Li, Y. Zhao, W. Wang, H. Zhang, and Z. Li, “Effects of prey refuges on a predator-prey model with a class of functional response: The role of refuges,” Mathematical Biosciences, vol. 218, no. 2, pp. 73–79, 2009. https://doi.org/10.1016/j.mbs.2008.12.008.

W. Ko and K. Ryu, “Two predator feeding on two prey species: A result on permanence,” Mathematical Biosciences, vol. 11, no. 2, pp. 246–252, 2010. http://dx.doi.org/10.1016/j.nonrwa.2008.10.056.

H. L. Li, L. Zhang, C. Hu, Y. L. Jiang, and Z. Teng, “Dynamical analysis of a functional order predator-prey model incorporting a prey refuge,” Journal of Applied Mathematics and computing, vol. 54, no. 1, pp. 246–252, 2017. https://doi.org/10.1007/s12190-016-1017-8.

S. Mondal and G. P. Samanta, “Dynamics of an additional food provided predator-prey system with prey refuge dependent on both species and constant harvest in predator,” Physica A: Statistical Mechanics and its Applications, vol. 534, no. 5, 2019. https://doi.org/10.1016/j.physa.2019.122301.

B. Mondal, S. Sarkar, P. K. Tiwari, and U. Ghosh, “A generalist predator-prey system with the effects of fear and refuge in deterministic and stochastic environments,” Mathematics and Computers in Simulation, vol. 225, no. 3, 2023. https://doi.org/10.1016/j.matcom.2023.09.022.

L. Dieci and J. Rebaza, “Point to periodic and periodic to periodic connection,” BIT Numerical Mathematics, vol. 44, no. 3, pp. 41–62, 2024. https://link.springer.com/article/10.1023/B:BITN.0000025093.38710.f6.

B. Leard, C. Lewis, and J. Rebaza, “Dynamics of ratio-dependent predator-prey models with non constant harvesting,” Discrete and Continuous Dynamical Systems, vol. 1, no. 2, pp. 435–449, 2017. https://doi.org/10.3934/dcdss.2008.1.303.

Authors

M. Krishika

Department of Mathematics, Annamalai University

krishika561999@gmail.com (Primary Contact)

S. Vijaya

Department of Mathematics, Annamalai University

Krishika, M., & Vijaya, S. (2026). Two Different Types Predator with Two Preys. Journal of the Indonesian Mathematical Society, 32(2), 2229. https://doi.org/10.22342/jims.v32i2.2229

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1. INTRODUCTION

Prey-predator interactions play a central role in ecological systems and form an important area of research in mathematical ecology. Various mathematical models have been developed to understand the dynamics of interacting species and the factors afecting their coexistence and stability [1,2].

Models involving multiple predator species have received considerable attention because they help explain competition and resource sharing mechanisms among predators. Tansky[3]investigated the efects of predator switching, while Hsu and Hubbell[4]analyzed competition between predators sharing common prey resources. Kirlinger [?] established conditions for the permanence of species in a two-predator two-prey system. Subsequently, Bhattacharyya and Mukhopadhyay[5]incorporated spatial migration and predator switching into prey-predator dynamics and Hadziabdic et al.[6]further examined the behavior of Lotka-Volterra systems with two predators and one prey.

Prey refuge is another important ecological mechanism that allows prey populations to avoid predation by utilizing protected habitats. Ma et al.[7]demonstrated that prey refuges significantly influence predator-prey interactions and functional responses. Ko and Ryu[8]showed that the incorporation of a constant prey refuge can enhance the stability of predator-prey systems. Li et al.[9]further analyzed a predator-prey model with prey refuge and discussed its efects on the long-term dynamics and stability of populations. Recent studies have also explored the combined efects of refuge and other ecological factors on predator-prey interactions [10,11].

Harvesting is another factor that strongly influences population dynamics. In natural and managed ecosystems, harvesting rates often vary with environmental conditions, seasonal changes and human activities. Dieci and Rebaza[12]investigated dynamical transitions in nonlinear systems, while Leard et al.[13]studied ratio-dependent predator-prey models with non-constant harvesting and demonstrated its impact on population stability and persistence.

Motivated by these studies, the present work proposes a prey-predator model consisting of two predator species, a prey refuge for the first prey population, and non-constant harvesting of the specialist predator. The objective is to investigate the dynamical behavior of the system and analyze the stability characteristics of the interacting populations.

2. MATHEMATICAL MODELING

The prey populations follow logistic growth and grow rapidly when small but slow down near their carrying capacities. They interact with predators based on prey availability. A prey refuge protects a fixed proportion \(\hat { m } k _ { 1 }\) of the population, leaving only \(( 1 - \hat { m } ) k _ { 1 }\) exposed to predation. The system includes a specialist predator \(k _ { 3 }\), which feeds on a limited prey type, and a generalist predator \(k _ { 4 }\), which consumes a wider range of prey. It also incorporates non-constant harvesting, where harvesting rates vary with time, population size, or environmental conditions. The corresponding mathematical formulation and its interaction represention are given below:

Table 1.Biological Description of the parameters for(1)
Parameters	Description
\(k_1\)	Prey population with refuge.
\(k_2\)	Prey population.
\(k_3\)	Specialized predator population.
\(k_4\)	Generalist predator population.
\(w_1, w_2\)	Intrinsic growth rate.
\(v_1, v_2\)	Carrying capacity.
\(l_1, l_2, l_3\)	Rate of predation.
\(a\)	Death rate of adult.
\(l_1, l_2\)	Rate of predation.
\(n_1, n_2, n_3\)	Natality rate of predators by each individual.
\(j\)	Half saturation constant.
\(c_1, c_2\)	Death rate of predator's
\(\hat{h}_1\)	Non constant harvesting rate.
\(\hat { m }\)	Refuge proportion \((0<\hat{m}\leq1).\)

Figure 1.Interaction structure of the prey predator model(1)

3. POSITIVE AND BOUNDEDNESS

The positivity and boundedness properties ensure that all species populations remain non-negative and finite over time. Ecologically, this reflects that populations cannot be negative and are limited by resources and environmental constraints, keeping the model biologically realistic.

Theorem 3.1.All solution of the system(1)initiating from\(R_{+}^{4}\)is positive for\(t ≥ 0\).

Proof.The first equation of(1)gives

\(k_1(t)=k_1(0)\exp\left(\int_0^t\left[w_1\left(1-\frac{k_1}{v_1}\right)-\frac{l_1(1-\hat{m})k_4}{k_1(1-\hat{m})+j}\right]dt\right)>0.\)

The second equation of the system(1)is

\(k_2(t)=k_2(0)\exp\left(\int_0^t\left[w_2\left(1-\frac{k_2}{v_2}\right)-\frac{l_2k_3}{k_2+j}-\frac{l_3k_4}{k_2+j}\right]dt\right)>0.\)

From the third equation of the system(1), we get

\(k_3(t)=k_3(0)\exp\left(\int_0^t\left[-c_1+\frac{n_1l_2k_2}{k_2+j}-\hat{h}_1\right]dt\right)>0.\)

The last equation of system(1)gives

\(k_4(t)=k_4(0)\exp\left(\int_0^t\left[-c_2+\frac{n_2l_2(1-\hat{m})k_1}{k_1(1-\hat{m})+j}+\frac{n_3l_3k_2}{k_2+j}\right]dt\right)>0.\)

This proves the theorem.

Theorem 3.2.All solutions of system(1)in\(R_{+}^{4}\)are uniformly bounded.

Proof.Let us define a function as follows:

Differentiate equation(1)with respect to \(’t’\), we get

\(\frac{d\bar{\omega}}{dt}=\frac{1}{\alpha}\frac{dk_1}{dt}+\frac{1}{\beta}\frac{dk_2}{dt}+\frac{1}{\gamma}\frac{dk_3}{dt}+\frac{1}{\delta}\frac{dk_4}{dt}\),

\(\frac{d\bar{\omega}}{dt}=\frac{1}{\alpha}\left[w_1-\frac{w_1k_1}{v_1}\right]+\frac{1}{\beta}\left[w_2-\frac{w_2k_2}{v_2}\right]-\frac{k_3}{\gamma}[c_1+\hat{h}_1]-\frac{k_4}{\delta}[c_2]\).

For any \(\xi >0\).

\(\frac{d\bar{\omega}}{dt}+\xi\bar{\omega}\leq\frac{v_1}{4\alpha w_1}(w_1+\xi)^2+\frac{v_2}{4\beta w_2}(w_2+\xi)^2\).

we define a constant \(η > 0\) such that \(\eta=\frac{\tilde{v}_1}{4\alpha w_1}(w_1+\xi)^2+\frac{v_2}{4\beta w_2}(w_2+\xi)^2\). It

shows that \(\frac{d\bar{\omega}}{dt}+\xi\bar{\omega}\leq\eta\). Applying theory of differential inequality, we obtain \(\bar{\omega}(k_1(t),k_2(t),k_3(t),k_4(t))<\frac{\eta}{\xi}\left(1-e^{-\xi t}\right)+\bar{\omega}(k_1(0),k_2(0),k_3(0),k_4(0))e^{-\xi t}\).

As \(t\rightarrow\infty\), this become \(0\leq\bar{\omega}(k_1(t),k_2(t),k_3(t),k_4(t))\leq\frac{\eta}{\xi}\). Hence, that concludes all solution of(1)with initial condition that initiate in \(R_{+}^{4}\) are bounded to the region \(\nabla=\left\{(k_1(t),k_2(t),k_3(t),k_4(t))\in R_{+}^{4}\mid\bar{\omega}=\frac{\eta}{\xi}+\epsilon\;\vee\;\epsilon\right\}\).

4. EXISTENCE OF EQUILIBRIUM POINTS

Various equilibrium points of the system have been examined. The system is observed to have six possible equilibrium points. \(\hat { m }\ne1\) and \(\hat { m }\ne1\), all the equilib-rium points satisfy the condition. These equilibrium points are now presented as follows:

(i) Trivial equilibrium \(\hat { E }_1 (0, 0, 0, 0)\) represents the complete extinction of all prey and predator populations under extreme ecological stress.

(ii) Coexistence of prey without predator equilibrium \(\hat { E }_2(v_1, v_2, 0, 0)\) represents the persistence of both prey species in the absence of predators.

(iii) Single prey with dual predator coexistence equilibrium

\(\hat{E}_3\left(\frac{c_2j}{(1-\hat{m})(n_2l_1-c_2)},0,jw_2-\frac{l_3}{l_2}\left[\frac{w_1n_2c_2j}{c_2}\right],\frac{w_1n_2c_2j}{c_2}\right)\)occurs when

\(n_2l_1>c_2\), \(jw_2>\frac{l_3w_1n_2\mathcal{J}}{l_2c_2}\)and \(1>\frac{c_2j}{v_1(1-\hat{m})(n_2l_1-c_2)}\)ensuring that one prey species persists and sustains both predator populations while the other prey is absent.

Where \(\mathcal{J} = 1-\frac{c_2j}{v_1(1-\hat{m})(n_2l_1-c_2)}\)

(iv) Generalized predator absent equilibrium\(\hat{E}_4\left(v_1,T,\frac{w_2j}{l_1v_1}\left[v_2(T+1)-T\right],0\right)\)

exists when \(n_1l_2>c_1+\hat{h}_1\)and\(v_2(T+1)>T\)under which both prey coexist with the specialist predator

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Two Different Types Predator with Two Preys

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