Predator Interference in a Predator–Prey Model with Mixed Functional and Numerical Responses

Alebraheem, Jawdat

doi:https://doi.org/10.1155/2023/4349573

Journal of Mathematics

On this page

Abstract Introduction Discussion and Conclusions Appendix Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 4349573 | https://doi.org/10.1155/2023/4349573

Predator Interference in a Predator–Prey Model with Mixed Functional and Numerical Responses

Jawdat Alebraheem¹

Academic Editor: Xiangfeng Yang

Received01 Sept 2022

Revised25 Nov 2022

Accepted26 Nov 2022

Published03 Jan 2023

Abstract

Predator interference refers to the competition effect on individuals of the same species through preventing or limiting access to a resource. In this paper, a novel predator–prey model that includes predator interference is considered. In the novel model, a high prey density is taken into consideration by investigating Crowley–Martin as a numerical response, while the growth rate of prey is logistically described and Holling type II is used as a functional response. In contrast to literature, this model has distinct and notable features: predator and prey species have an intraspecific competition, and this model has a numerical response that generalizes several of the commonly utilized depictions of predator interference. The boundedness of the model is proved. Predator interference effects are theoretically and numerically studied in terms of stability, coexistence, and extinction. Theoretical results show that the system has no effect on coexistence or extinction in these types of models. Numerical simulations present two different dynamics with identical results on the theoretical side of model stabilization for both dynamics. Moreover, the numerical simulations show that there is a change from periodic to steady state dynamics as the value of predator interference increases. This means that the coexistence probability of the model increases. The attained results are explained from both a mathematical and an ecological points of view.

1. Introduction

Recent studies have used differential equations to describe a variety of applications, such as [1–8]. Population models, a central topic in many sciences due to their significance, study population change over time. One of the first scientists who noticed geometric population growth was Malthus [9], whose model is known by his name and is called the exponential model. After that, Verhulst [10] formulated the logistic model for population growth, which is called the Verhulst model and takes the sigmoidal form. The predator–prey model is one of the most common types of population models used to describe multispecies interactions. Lotka [11] and Volterra [12] independently founded the Lotka–Volterra model, which is considered the fundamental building block of predator–prey interactions and consists of two nonlinear differential equations. In order to make the Lotka–Volterra model more realistic, many ecological processes have been taken into account.

Two of the main components are functional and numerical responses, which connect the equations of predator and prey [13]. There are different kinds of functional and numerical responses. They are classified depending on the presence of predators and prey densities in the formulation of functional and numerical responses [14, 15]. Holling type II is one of the most commonly used types of functional and numerical responses and has been widely used in many studies with different ideas, for example [16–21]. Generally, the main findings of these studies addressed the dynamics of the Holling type II models only; they derived their conditions, obtained their theoretical results, and performed simulations that followed certain dynamics such as stability, coexistence, and extinction under some external effects such as allee and refuge effects.

Holling type II is classified as prey-dependent functional and numerical responses because it is subjected to prey density only [22]. However, the Holling type II type was developed by including terms in the denominator to represent predator interference, and thus these types become dependent on the density of prey and predator as in the Beddington–DeAngelis (BD) [23, 24] and Crowley–Martin (CM) [25] types, but the difference between them is that in the case of the Crowley–Martin type, predator interference is taken into account even in a high prey density, whereas in the type of Beddington–DeAngelis, it excludes predator interference in a high prey density. Recent studies, such as those [26, 27], have used the Crowley–Martin type to discuss the dynamic behaviors of prey and predator species. They obtained the stability, coexistence, and extinction conditions of these models.

Predator–prey systems with mixed functional and numerical responses have been used in a few studies to describe some kinds of environment, which study the dynamic behaviors of these models such as [28–30]. The authors of [28] used the Holling type II and Beddington–DeAngelis type as mixed functional responses to investigate the stability and find out the persistence condition; while in [29], they used the Holling type IV and Holling type I as mixed functional responses to examine the local and global stability and obtain the persistence condition. Finally, in [30], the authors investigated the stability of the equilibrium points of a discrete Lotka–Volterra predator–prey model with mixed functional responses of Holling type I and II, and the model was studied in terms of its long-term behavior when each control parameter was changed.

However, Holling type II is widely used and well documented in many theoretical and experimental studies; Crowley–Martin was developed based on Holling type II. To the best of our knowledge, Holling type II and Crowley–Martin have not been used as mixed functional responses to investigate the dynamics. This motivates us to investigate the dynamics of the predator–prey model that incorporates mixed functional responses from Crowley–Martin and Holling type II, respectively. We also examine how predator inference affects the dynamic behaviors of this model.

Interference predator is a concept that describes the effect of competition on individuals of the same species through preventing or limiting access to a resource. It is considered one of the intraspecific competition mechanisms [31]. The dynamics of some mathematical models that contain intraspecific competition have been discussed in many research papers [26–46], which plays a significant role in systems’ dynamics. Although in both laboratory and natural systems, predator interference has been observed quite frequently, its effects on predator–prey dynamics have only been studied in a limited number of studies that have concentrated on the use of Beddington–DeAngelis as functional and numerical responses [47–54], but Beddington–DeAngelis excludes predator interference at high prey densities.

The main contribution of this research is the study of the interference of predators in a comprehensive approach, including a high prey density, which is taken into consideration through an investigation of Crowley–Martin as a numerical response, while the growth rate of prey is logistically described and Holling type II is used as a functional response. This model has remarkable features: the prey and predator species have an intraspecific competition aspect, and this model’s numerical response is considered the most generalized of the commonly utilized depictions of predator interference. This study examines interference predator effects on stability, coexistence, and extinction of this model.

This manuscript is organized as follows: Section 2 introduces the model’s formulation and the proof of boundedness. Then, Kolmogorov conditions are discussed in Section 3. In Section 4, stability analysis and interference predator effects are explained theoretically. Then, numerical simulations are executed to support the theoretical results in Section 5. Finally, conclusions and discussion are given in Section 6.

2. Formulation and Boundedness of the Model

In this section, we introduce a continuous-time model that describes the interactions of predator and prey species and describes the growth rate of prey logistically in the absence of predator using mixed functional and numerical responses. The model is constructed as follows:

With and that represent positive initial conditions. indicates the growth rate of prey and symbolizes the carrying capacity of the prey in the absence of predation. is the accessibility parameter of prey by a predator, is the death rate of the predator, is conversion rate into biomass of the predator, is the handling time and represents the interference predator.

Theorem 1. The solutions on of model (1) in is positive and bounded with any given positive initial values.

Proof. We assume with positive initial conditions is any solution of model (1), so there is a unique solution . We take and observe that where . We consider .
The derivative of the function is as follows:The denominator of equation (1b) can be written as follows:Equation (3) can be written as follows:After simplification of equation (5), then we get the following equation:The inequality (11) obtains the following equation:Consequently, and are ultimately bounded.
We obtain that model (1) has three non-negative equilibrium points that are , , and . and exist without conditions. The existence of coexistence point is gotten by obtaining the positive real root of the following system algebraic equations:To discuss the existence of the coexistence point , we introduce the following analysis:
Through equation (13a), we assume as follows:One gets the following:(i)When , then . As a result, has no conditions attached to it.(ii)When , then . As a result, has no conditions attached to it. is always less than zero because , and the parameters are positive, so .
According to the above analysis, isocline (14) passes through points and , and is a decreasing function of .
Through equation (13b), supposeOne gets the following:(i)When , then . It is clear that under the following condition (17):(ii)As a result, the slope of isocline (16) always remains positive and it passes through the point under condition (17).
It is concluded from the above analysis that the two isoclines (14) and (16) intersect at a unique point when

Proposition 1. System (1) has a unique positive equilibrium point if conditions (17) and (18) are satisfied.

3. Kolmogorov Conditions

In this section, Kolmogorov conditions are applied to ensure that there is a unique nontrivial equilibrium point for model (1). In addition, Kolmogorov conditions are used to obtain the persistence and extinction conditions and validate the model.

Thus, model (1) is written as follows:

The first set of Kolmogorov conditions states that the growth rate of prey and predators decreases when the number of prey is fixed and the number of predators increases. The conditions are expressed as follows:

and are negative since all the variables and parameters are positive, so the growth rate of prey and predators decreases.

The second condition is that the model has a carrying capacity. The condition is indicated as follows:

The third condition is that if the model has a small predator population, it will include a minimum prey value.

The fourth condition is that the prey coexists with predators if the following condition holds:

This implies that

But, if condition (25) is violated, then the predator becomes extinct,

From Kolmogorov analysis, we attain the following theorems:

Proposition 2. Under condition (25), prey coexists with predators.

Corollary 1. The predators will become extinct if condition (26) is satisfied.

Through conditions (25) and (26), we conclude the following corollary:

Corollary 2. Predator interference does not affect coexistence and extinction.

4. Stability Analysis

For population dynamics, the qualitative analysis of differential equations as stability plays an influential role in following the dynamic behaviors of populations [55]. The stability analysis of non-negative equilibrium points is discussed through the following theorems.

We get the Jacobian matrix of model (1) as follows:

The following theorems are obtained to study the stability of positive equilibrium points.

Theorem 2. in model (1) is a saddle point (i.e., unstable).

Proof. By substituting , the Jacobian matrix of the model (1) is as follows:It is noticed that . Therefore, is a saddle point.

Theorem 3. Suppose that holds, then the equilibrium point is stable.

Proof. By substituting , the Jacobian matrix of model (1) is as follows:It has been observed that , so and if and only if . Consequently, is stable ifWe select the values of the parameters and initial conditions, which are as follows:
, , , , and , to plot two of equilibrium points and , while the third equilibrium point, which represents the coexistence equilibrium point , will be considered in detailed in the next section. is a saddle point, as theoretically proved in Theorem 2 in all cases, it is shown in Figure 1 and Figure 2 that the stream of arrows is moving away from point . With the values of parameters, the equilibrium point is represented by , which has two cases as proved theoretically in Theorem 3, in Figure 1, it is observed that is stable, as it heads towards the point , because condition (30) is satisfied in Theorem 3. However, if the value of the death rate of the predator (s) is changed, then condition (30) is not satisfied and point becomes a saddle point (i.e., an unstable case) as shown in Figure 2, where the stream of arrows is moving away from the point .

Theorem 4. The coexistence equilibrium point is locally asymptotically stable if it holds the following conditions:

Proof. The Jacobian matrix of is as follows:The determinant of is as follows:By simplifying the previous determinant, we get the following equation:Thus, if and .
Then, we obtain a trace of as follows:Therefore, if and .
This implies that the coexistence equilibrium point is stable.

Theorem 5. Model (1) has a periodic solution in under the following condition:

Proof. Let,By multiplying the Dulac function by and , we get the following equation:By taking the divertive of according to we have the following equation:Also for we have the following equation:By taking the divertive of according to we have the following equation:As observed, there is a change in the signs ofBy Bendixson–Dulac criterion, model (1) has a periodic solution under condition (37).

5. Numerical Simulations

By selecting parameter values that are biologically feasible, numerical simulations are conducted to present the dynamics of model (1), which show two different dynamics, steady state and periodic dynamics, as obtained in Section 4. In both dynamics, the values of parameters are selected so that the predator and prey coexist, which satisfies Proposition 2. In addition, interference predator effects are explained in terms of the dynamics of predator–prey interactions. Time series and phase portraits with nullclines figures are used to explain these effects and show the dynamics.

The first case is steady state dynamics:

, , , , , , , and . However, the value of the interference predator changes for each figure.

Second case is periodic dynamics:

, , , , , , , , and . However, the value of the interference predator (ν) changes for each figure.

Figures 3–6 present steady state dynamics, and with increasing the predator interference value for each figure, it is noticed that the dynamic behavior is still in a steady state, but the densities of prey and predator are affected, where the density of prey increases and the density of predator decreases at high values of predator interference, as explained by the intersection point of nullclines. Coexistence and extinction are not affected by predator interference. This corresponds with theoretical results (i.e., Corollary 3.3). In addition, it is clear that nullclines (8) intersect uniquely; this shows that the coexistence equilibrium point is unique.

In the periodic dynamic (Figure 7), it is observed that the size of the oscillation is reduced, as shown in Figure 8, by increasing the value of predator interference. However, as the predator interference value increases, the dynamics change to a steady state, as shown in Figures 9 and 10. This makes the model more stable, in addition to changing the densities of predator and prey as in a steady state dynamic. The change from periodic to steady state dynamic involves the coexistence probability increasing and the extinction probability decreasing as the value of predator interference increases.

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

6. Discussion and Conclusions

In this paper, Holling type II and Crowley–Martin functional and numerical responses are combined to form a reasonable predator–prey model. This model has remarkable features; the prey and predator species have an intraspecific competition aspect, and this model’s numerical response is considered the most generalized of the commonly utilized depictions of predator interference. Interference predators are included as an influential feature in this model that may affect the dynamics of predator–prey interactions, so this study aims to present the dynamics of this model. In addition, Crowley–Martin was developed based on Holling type II, and it has been widely used in many experimental and theoretical studies in the literature. The boundedness of this model is proved by Theorem 1.

Through Kolmogorov conditions, we conclude some significant results as follows:(i)Based on Kolmogorov’s four conditions (20–26), respectively, predator–prey model (1) is endorsed.(ii)Ensure that there is a unique nontrivial equilibrium point for model (1), as illustrated in Proposition 1.(iii)Attain the coexistence and extinction conditions, which are conditions (25) and (26).(iv)Corollary 3.4 is an important result, which shows that the interference predator has no effect on the coexistence and extinction in this kind of models.

Stability analysis of model (1) is introduced to obtain the stability conditions as shown through Theorems 2–4. According to Theorem 5, model (1) has a periodic solution, so model (1) has two different dynamics: steady state and periodic dynamics.

The numerical simulations present two different dynamics: steady state and periodic dynamics, and the effect of predator interference is shown on the dynamics of predator–prey interactions. The numerical outcomes of predator interference effects are summarized as follows:(i)In the steady state dynamics, there is an increase in the density of prey and a decrease in the density of predators, which can be interpreted as the presence of aggressive behavior by one or more individuals, which affects the rest of the predators [28]. Thus, this means that predator interference has no effect on coexistence or extinction, not even the stability of the system.(ii)In the periodic dynamic, the effect is clear through the changing of the dynamics by making the predator–prey system stabilize with increasing predator interference in this kind of models. This result is consistent with the literature such as [47–51].(iii)In addition, a significant result is obtained through the change from periodic to steady state dynamics. This can be interpreted as a meaning that the probability of coexistence increases and the extinction probability decreases in the model when the value of predator interference is increased.

Appendix

Proof. As model (1) is Kolmogorov type that satisfies the Kolmogorov conditions, so Freedman [56] proved that these kinds of systems has a unique nontrivial equilibrium point as referred in proof of a Kolmogorov type theorem (Theorem 5.1), he used the figures and the Poincare–Bendixson theorem to explain his proof. Proposition 2 Under condition (25), prey coexists with predators. Proof. Through Kolmogorov type theorem (Theorem 5.1), which model (1) is Kolmogorov type that satisfies the Kolmogorov conditions, coexistence condition is one of these conditions as referred in proof of a Kolmogorov type theorem (Theorem 5.1), he used the figures and the Poincare–Bendixson theorem to explain his proof. Corollary 1 The predators will become extinct, if the condition (26) is satisfied. Proof. It can show through the analysis of Preposition 1 that the same condition (18) presents when and . Also, Through Kolmogorov type theorem (Theorem 5.1), which model (1) is Kolmogorov type that satisfies the Kolmogorov conditions, the extinction condition is one of these conditions as referred in proof of a Kolmogorov type theorem (Theorem 5.1). Corollary 2 Predator interference does not affect coexistence and extinction. Proof. Since there is no interference predator parameter () in the coexistence and extinction conditions (25) and (26), respectively, then complete the proof of this corollary.

Data Availability

All data generated or analyzed during this study are included in this article.

Conflicts of Interest

The author declares no conflicts of interest.

Acknowledgments

The author extends his appreciation to the deputyship for Research & Innovation, Ministry of Education in Saudi Arabia for funding this research work through the project number (IFP-2020-98).

References

C. López Zazueta, O. Bernard, and J.-L. Gouzé, “Analytical reduction of nonlinear metabolic networks accounting for dynamics in enzymatic reactions,” Complexity, vol. 2018, 22 pages, 2018.
View at: Publisher Site | Google Scholar
J. Alebraheem, “Paradox of enrichment in a stochastic predator-prey model,” Journal of Mathematics, vol. 2020, 8 pages, 2020.
View at: Publisher Site | Google Scholar
T. Jin and X. Yang, “Monotonicity theorem for the uncertain fractional differential equation and application to uncertain financial market,” Mathematics and Computers in Simulation, vol. 190, pp. 203–221, 2021.
View at: Publisher Site | Google Scholar
S. Feng, Z. Feng, C. Ling, C. Chang, and Z. Feng, “Prediction of the covid-19 epidemic trends based on seir and ai models,” PLoS One, vol. 16, no. 1, Article ID e0245101, 2021.
View at: Publisher Site | Google Scholar
N. Goyal, P. Kulczycki, and M. Ram, Eds.in Differential Equations in Engineering: Research and Applications, CRC Press, Boca Raton, FA, USA, 1st edition, 2021.
View at: Publisher Site
T. Jin, X. Yang, H. Xia, and H. Ding, “Reliability index and option pricing formulas of the first-hitting time model based on the uncertain fractional-order differential equation with caputo type,” Fractals, vol. 29, no. 01, pp. 2150012–2150183, 2021.
View at: Publisher Site | Google Scholar
I. F. Barna and L. Mátyás, “Advanced analytic self-similar solutions of regular and irregular diffusion equations,” Mathematics, vol. 10, no. 18, p. 3281, 2022.
View at: Publisher Site | Google Scholar
C. Tian, T. Jin, X. Yang, and Q. Liu, “Reliability analysis of the uncertain heat conduction model,” Computers & Mathematics with Applications, vol. 119, pp. 131–140, 2022.
View at: Publisher Site | Google Scholar
T. R. Malthus, An Essay on the Principle of Population, as it affects the future improvement of society. With Remarks on the Speculations of Mr Godwin, M. Condorcet, and other writers, Kelley, Bloomington, Indiana, 7th edition, 1798, Harmondsworth: Penguin.
P.-F. Verhulst, “Mathematical researches into the law of population growth increase,” Nouv.Mem.Acad.R.Sci.Bruxelles, vol. 18, pp. 14–54, 1845.
View at: Google Scholar
A. J. Lotka, Elements of Physical Biology, Williams & Wilkins, Baltimore, MD, USA, 1925.
V. Volterra, “Fluctuations in the abundance of a species considered mathematically,” Nature, vol. 118, no. 2972, pp. 558–560, 1926.
View at: Publisher Site | Google Scholar
L. L. Rockwood, Introduction to Population Ecology, Cambridge Univ Press, Cambridge, UK, 2006.
G. T. Skalski and J. F. Gilliam, “Functional responses with predator interference: viable alternatives to the holling type II model,” Ecology, vol. 82, pp. 3083–3092, 2001.
View at: Publisher Site | Google Scholar
Y. Abu-Hasan and J. Alebraheem, “Functional and numerical response in prey-predator system,” in Proceedings of the AIP Conference, vol. 1651, p. 3, Ekaterinburg, Russia, May 2015.
View at: Google Scholar
J. Alebraheem and Y. Abu-Hasan, “Persistence of predators in a two predators-one prey model with non-periodic solution,” Applied Mathematical Sciences, vol. 6, pp. 943–956, 2012.
View at: Google Scholar
Q. Yue, “Dynamics of a modified Leslie–Gower predator–prey model with Holling-type II schemes and a prey refuge,” SpringerPlus, vol. 5, no. 1, p. 461, 2016.
View at: Publisher Site | Google Scholar
J. Alebraheem, “Relationship between the paradox of enrichment and the dynamics of persistence and extinction in prey-predator systems,” Symmetry, vol. 10, p. 532, 2018.
View at: Publisher Site | Google Scholar
H. Molla, M. Sabiar Rahman, and S. Sarwardi, “Dynamics of a predator–prey model with holling type II functional response incorporating a prey refuge depending on both the species,” International Journal of Nonlinear Sciences and Numerical Stimulation, vol. 20, no. 1, pp. 89–104, 2019.
View at: Publisher Site | Google Scholar
S. Wang, H. Yu, C. Dai, and M. Zhao, “The dynamical behavior of a predator-prey system with holling type II functional response and allee effect,” Applied Mathematics, vol. 11, no. 05, pp. 407–425, 2020.
View at: Publisher Site | Google Scholar
Y. Wu, F. Chen, and C. Du, “Dynamic behaviors of a nonautonomous predator–prey system with Holling type II schemes and a prey refuge,” Advances in Difference Equations, vol. 2021, no. 1, p. 62, 2021.
View at: Publisher Site | Google Scholar
C. S. Holling, “Some characteristics of simple types of predation and parasitism,” The Canadian Entomologist, vol. 91, no. 7, pp. 385–398, 1959.
View at: Publisher Site | Google Scholar
J. R. Beddington, “Mutual interference between parasites or predators and its effect on searching efficiency,” Journal of Animal Ecology, vol. 44, no. 1, pp. 331–340, 1975.
View at: Publisher Site | Google Scholar
D. L. DeAngelis, R. A. Goldstein, and R. V. ’O’Neill, “A model for tropic interaction,” Ecology, vol. 56, no. 4, pp. 881–892, 1975.
View at: Publisher Site | Google Scholar
P. H. Crowley and E. K. Martin, “Functional responses and interference within and between year classes of a dragonfly population,” Journal of the North American Benthological Society, vol. 8, no. 3, pp. 211–221, 1989.
View at: Publisher Site | Google Scholar
X. Y. Shi, X. Y. Zhou, and X. Y. Song, “Analysis of a stage-structured predator–prey model with Crowley–Martin function,” J. Appl. Math. Comput.., vol. 36, no. 1-2, pp. 459–472, 2011.
View at: Publisher Site | Google Scholar
J. Alebraheem, “Dynamics of a predator–prey model with the effect of oscillation of immigration of the prey,” Diversity, vol. 13, no. 1, p. 23, 2021.
View at: Publisher Site | Google Scholar
H. Baek and D. Kim, “Dynamics of a predator-prey system with mixed functional responses,” Journal of Applied Mathematics, vol. 2014, Article ID 536019, 10 pages, 2014.
View at: Publisher Site | Google Scholar
V. Madhusudanan and S. Vijaya, “Stability analysis in prey–predator system with mixed functional response,” World Journal of Engineering, vol. 13, no. 4, pp. 364–369, 2016.
View at: Publisher Site | Google Scholar
M. F. Elettreby, A. Khawagi, and T. Nabil, “Dynamics of a discrete prey-predator model with mixed functional response,” International Journal of Bifurcation and Chaos, vol. 29, no. 14, Article ID 1950199, 2019.
View at: Publisher Site | Google Scholar
O. Gilad, “Competition and Competition Models,” Encyclopedia of Ecology, S. E. Jørgensen, D. F. Brian, Eds. Academic Press: Cambridge, MA, USA, 2008.
View at: Publisher Site | Google Scholar
J. Alebraheem and Y. Abu-Hassan, “Simulation of complex dynamical behaviour in prey predator model,” in Proceedings of the 2012 International Conference on Statistics in Science, Business and Engineering, Langkawi, Malaysia, September 2012.
View at: Publisher Site | Google Scholar
X. Y. Meng, H. F. Huo, H. Xiang, and Q. Y. Yin, “Stability in a predator-prey model with Crowley-Martin function and stage structure for prey,” Applied Mathematics and Computation, vol. 232, pp. 810–819, 2014.
View at: Publisher Site | Google Scholar
A. P. Maiti, B. Dubey, and A. Chakraborty, “Global analysis of a delayed stage structure prey–predator model with Crowley–Martin type functional response,” Mathematics and Computers in Simulation, vol. 162, pp. 58–84, 2019.
View at: Publisher Site | Google Scholar
J. P. Tripathi, S. Bugalia, V. Tiwari, and Y. Kang, “A predator–prey model with Crowley Martin functional response: a nonautonomous study,” Natural Resource Modeling, vol. 33, no. 4, Article ID e12287, 2020.
View at: Publisher Site | Google Scholar
P. Panja, “Dynamics of a predator–prey model with crowley–martin functional response, refuge on predator and harvesting of super-predator,” Journal of Biological Systems, vol. 29, no. 3, pp. 631–646, 2021.
View at: Publisher Site | Google Scholar
B. Dubey and R. K. Upadhyay, “Persistence and extinction of one-prey and two-predators system,” Nonlinear Analysis Modelling and Control, vol. 9, no. 4, pp. 307–329, 2004.
View at: Publisher Site | Google Scholar
S. Gakkhar, B. Singh, and R. K. Naji, “Dynamical behavior of two predators competing over a single prey,” Biosystems, vol. 90, no. 3, pp. 808–817, 2007.
View at: Publisher Site | Google Scholar
X. Tian and R. Xu, “Global dynamics of a predator-prey system with holling type II functional response,” Nonlinear Analysis Modelling and Control, vol. 16, no. 2, pp. 242–253, 2011.
View at: Publisher Site | Google Scholar
J. Alebraheem and Y. A. Hasan, “Dynamics of a two predator–one prey system,” Computational and Applied Mathematics, vol. 33, no. 3, pp. 767–780, 2013.
View at: Publisher Site | Google Scholar
D. Mukherjee, “The effect of refuge and immigration in a predator–prey system in the presence of a competitor for the prey,” Nonlinear Analysis: Real World Applications, vol. 31, pp. 277–287, 2016.
View at: Publisher Site | Google Scholar
H. Berestycki and A. Zilio, “Predator–prey models with competition: the emergence of territoriality,” The American Naturalist, vol. 193, no. 3, pp. 436–446, 2019.
View at: Publisher Site | Google Scholar
H. Deng, F. Chen, Z. Zhu, and Z. Li, “Dynamic behaviors of Lotka–Volterra predator–prey model incorporating predator cannibalism,” Advances in Difference Equations, vol. 2019, no. 1, p. 359, 2019.
View at: Publisher Site | Google Scholar
Z. Zhu, R. Wu, L. Lai, and X. Yu, “The influence of fear effect to the Lotka–Volterra predator–prey system with predator has other food resource,” Advances in Difference Equations, vol. 2020, no. 1, p. 237, 2020.
View at: Publisher Site | Google Scholar
P. Zheng, “On a two-species competitive predator-prey system with density-dependent diffusion,” Mathematical Biosciences and Engineering, vol. 19, no. 12, pp. 13421–13457, 2022.
View at: Publisher Site | Google Scholar
Y. Long, L. Wang, and J. Li, “Uniform persistence and multistability in a two-predator–one-prey system with inter-specific and intra-specific competition,” International Journal of Applied Mathematics and Computer Science., vol. 68, no. 2, pp. 767–794, 2021.
View at: Publisher Site | Google Scholar
G. D. Ruxton, W. S. C. Gurney, and A. De Roos, “Interference and generation cycles,” Theoretical Population Biology, vol. 42, no. 3, pp. 235–253, 1992.
View at: Publisher Site | Google Scholar
G. Huisman and R. J. DeBoer, “A formal derivation of the “Beddington” functional response,” Journal of Theoretical Biology, vol. 185, no. 3, pp. 389–400, 1997.
View at: Publisher Site | Google Scholar
R. Arditi, J. M. Callois, Y. Tyutyunov, and C. Jost, “Does mutual interference always stabilize predator-prey dynamics? A comparison of models,” Comptes Rendus Biologies, vol. 327, no. 11, pp. 1037–1057, 2004.
View at: Publisher Site | Google Scholar
B. S. R. V. Prasad, M. Banerjee, and P. D. N. Srinivasu, “Dynamics of additional food provided predator-prey system with mutually interfering predators,” Mathematical Biosciences, vol. 246, no. 1, pp. 176–190, 2013.
View at: Publisher Site | Google Scholar
L. Přibylová and L. Berec, “Predator interference and stability of predator–prey dynamics,” Journal of Mathematical Biology, vol. 71, no. 2, pp. 301–323, 2015.
View at: Publisher Site | Google Scholar
A. Jana and S. Kumar Roy, “Holling-Tanner prey-predator model with Beddington-DeAngelis functional response including delay,” International Journal of Modelling and Simulation, vol. 42, no. 1, pp. 86–100, 2022.
View at: Publisher Site | Google Scholar
Y. Shao and W. Kong, “A predator–prey model with beddington–DeAngelis functional response and multiple delays in deterministic and stochastic environments,” Mathematics, vol. 10, no. 18, p. 3378, 2022.
View at: Publisher Site | Google Scholar
L. N. Guin, R. Murmu, H. Baek, and K.-H. Kim, “Dynamical analysis of a beddington–DeAngelis interacting species system with prey harvesting,” Mathematical Problems in Engineering, vol. 2020, Article ID 7596394, 22 pages, 2020.
View at: Publisher Site | Google Scholar
C. Tunç and O. Tunç, “Qualitative analysis for a variable delay system of differential equations of second order,” Journal of Taibah University for Science, vol. 13, no. 1, pp. 468–477, 2019.
View at: Publisher Site | Google Scholar
I. Freedman, Deterministic Mathematical Models in Population Ecology, Marcel Dekker, New York, NY, USA, 1980.

Copyright

Copyright © 2023 Jawdat Alebraheem. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

571

Downloads

532

Citations