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

Authors: Jawdat Alebraheem
Journal: Journal of Mathematics
 (2023)

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## 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.


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## Body

## 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: Section2 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:(1a)dXdt=rX1−Xk−λXY1+ξX,(1b)dYdt=−sY+λβXY1+ξX+νY+ξνXY.WithX0=X0 and Y0=Y0 that represent positive initial conditions. r indicates the growth rate of prey and k symbolizes the carrying capacity of the prey in the absence of predation. λ is the accessibility parameter of prey by a predator, s is the death rate of the predator, β is conversion rate into biomass of the predator, ξ is the handling time and ν represents the interference predator.Theorem 1.
The solutions ont≥0 of model (1) inR+2 is positive and bounded with any given positive initial values.Proof.
We assumeXt,Yt with positive initial conditions is any solution of model (1), so there is a unique solution Xt,Yt∈R+2. We take ψ:=min1,s and observe that rX1−X/k+X≤Γ where Γ=k+2kr+kr2/4r. We consider Ft=Xt+1/βYt.
The derivative of the functionF is as follows:(2)dFdt=dXdt+1βdYdt,(3)dFdt=r1−Xk−λY1+ξXX+−sβ+λX1+ξX+vy+ξvXYY.
The denominator of equation (1b) can be written as follows:(4)1+ξX+νY+ξνXY=1+ξX1+vY.
Equation (3) can be written as follows:(5)=rX1−Xk−λYX1+vY1+ξX+νY+ξνXY+−sβY+λXY1+ξX+vy+ξvXY.
After simplification of equation (5), then we get the following equation:(6)=rX1−Xk−sβY−λvXY21+ξX+vy+ξvXY,(7)≤rX1−Xk−sβY,(8)=rX1−Xk+X−X−sβY,(9)≤rX1−Xk+X−X+sβY(10)≤rX1−Xk+X−ψF,(11)≤Γ−ψF.
The inequality (11) obtains the following equation:(12)Ft≤Γψ+F0e−ψt≤maxΓψ,F0.
Consequently,Xt and Yt are ultimately bounded.
We obtain that model (1) has three non-negative equilibrium points that areE00,0, E1k,0, and E2X…,Y…. E0 and E1 exist without conditions. The existence of coexistence point E2X…,Y… is gotten by obtaining the positive real root of the following system algebraic equations:(13a)r1−Xk−λY1+ξX=0,(13b)−s+λβX1+ξX+νY+ξνXY=0.
To discuss the existence of the coexistence pointE2X…,Y…, we introduce the following analysis:
Through equation (13a), we assume as follows:(14)UX…,Y…=r1−X…k−λY…1+ξX…=0.
One gets the following:(i)
WhenX…=0, then Y…=r/λ=Y…i. As a result, Y…i>0 has no conditions attached to it.(ii)
WhenY…=0, then X…=k=X…i. As a result, X…i>0 has no conditions attached to it.(15)dY…dX…=−∂U/∂X…∂U/∂Y…=−−r/k−ξλY…/1+ξX…2−λ/1+ξX…=−r/k−ξλY…/1+ξX…2λ/1+ξX…,∴dY…dX…=−rk1+ξX…2−ξλY….
−r/k1+ξX…2−ξλY… is always less than zero because X…, Y… and the parameters are positive, so dY…/dX…<0.
According to the above analysis, isocline (14) passes through points X…i,0 and 0,Y…i, and Y… is a decreasing function of X….
Through equation (13b), suppose(16)VX…,Y…=−s+λβX…1+ξX…+νY…+ξνX…Y…=0.
One gets the following:(i)
WhenY…=0, then X…=S/λβ−Sξ=X…ii. It is clear that X…ii>0 under the following condition (17):(17)λβ>Sξ,(ii)
dY…/dX…=−∂V/∂X…/∂V/∂Y…=1+νY…/vX…1+ξX…>0.
As a result, the slope of isocline (16) always remains positive and it passes through the point X…ii,0 under condition (17).
It is concluded from the above analysis that the two isoclines (14) and (16) intersect at a unique point when(18)X…ii<X…i,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:(19a)dXdt=rX1−Xk−λXY1+ξX=XUX,Y(19b)dYdt=−sY+λβXY1+ξX+νY+ξνXY=YVX,Y.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:(20)∂U∂Y=−λ1+ξX<0,(21)∂V∂Y=−λβX1+ξX+νY+ξνxY2<0.∂U/∂Y and ∂V/∂Y 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:(22)∃J>0∋UJ,0=0,thenJ=k.The third condition is that if the model has a small predator population, it will include a minimum prey value.(23)∃ψ>0∋Vψ,0=0,thenψ=λβ−sξs.The fourth condition is that the prey coexists with predators if the following condition holds:(24)J>ψ.This implies that(25)k>λβ−sξs.But, if condition (25) is violated, then the predator becomes extinct,(26)k≤λβ−sξs.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:(27)JX,Y=r−2rXk−λY1+ξX2−λX1+ξXλβY1+νY1+ξX+νY+ξνxY2−s+λβX1+ξX1+ξX+νY+ξνxY2.The following theorems are obtained to study the stability of positive equilibrium points.Theorem 2.
E00,0 in model (1) is a saddle point (i.e., unstable).Proof.
By substitutingX=Y=0, the Jacobian matrix of the model (1) is as follows:(28)J0,0=r00−s.
It is noticed thatJ0,0=−rs<0. Therefore, E00,0 is a saddle point.Theorem 3.
Suppose thats>λβk/1+ξk holds, then the equilibrium point E3k,0 is stable.Proof.
By substitutingX=kandY=0, the Jacobian matrix of model (1) is as follows:(29)JΥ,0=−r−λk1+ξk0−s+λβk1+ξk.
It has been observed thatJk,0=rs−rλβk/1+ξk, so Jk,0>0 and trJk,0<0 if and only if s>λβk/1+ξk. Consequently, E3k,0 is stable if(30)s>λβk1+ξk.
We select the values of the parameters and initial conditions, which are as follows:
r=2, k=4, λ=2.0, ξ=3.0, and v=0.9, to plot two of equilibrium points E00,0 and E1k,0, while the third equilibrium point, which represents the coexistence equilibrium point E2X…,Y…, will be considered in detailed in the next section. E00,0 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 E00,0. With the values of parameters, the equilibrium point E1k,0 is represented by E14,0, which has two cases as proved theoretically in Theorem 3, in Figure 1, it is observed that E14,0 is stable, as it heads towards the point E14,0, 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 E14,0 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 E14,0.Figure 1
A stream of arrows shows the saddle point ofE00,0 and stable point of E14,0.Figure 2
A stream of arrows shows the saddle point ofE00,0 and saddle point of E14,0.Theorem 4.
The coexistence equilibrium pointEX…,Y… is locally asymptotically stable if it holds the following conditions:(31)2X…>k,(32)Y…>λβX…1+ξX…1+νY…2.Proof.
The Jacobian matrix ofEX…,Y… is as follows:(33)JX…,Y…=r−2rX…k−λY…1+ξX…2−λX…1+ξX…λβY…1+νY…1+ξX…+νY…+ξνX…Y…2−s+λβX…1+ξX…1+ξX…+νY…+ξνX…Y…2.
The determinant ofJX^,Y^ is as follows:(34)JX^,Y^=−rs+rλβX…1+ξX…+νY…+ξνX…Y…2+2rsX…k−2rλβX…21+ξX…k1+ξX…+νY…+ξνX…Y…2+sλY…1+ξX…2−λ2βX…Y…1+ξX…1+ξX…21+ξX…+νY…+ξνX…Y…2.
By simplifying the previous determinant, we get the following equation:(35)JX^,Y^=rs2X…k−1+λβX…1+ξX…+νY…+ξνX…Y…2r1+ξX…−2rX…k−λY…1+ξX…+sλY…1+ξX…2.
Thus,JX…,Y…>0 if 2X…>k and 1+ξX…>2rX…/k+λY…/1+ξX….
Then, we obtain a trace ofJX…,Y… as follows:(36)traceJX…,Y…=r−2rX…k−λY…1+ξX…2−s+λβX…1+ξX…1+ξX…+νY…+ξνX…Y…2,traceJX…,Y^=−r2X…k−1−s−λ1+ξX…Y…−λβX…1+νY…2.
Therefore,traceJX…,Y…<0 if 2X…>k and Y…>λβX…/1+νY…2.
This implies that the coexistence equilibrium pointEX…,Y… is stable.Theorem 5.
Model (1) has a periodic solution inR+2 under the following condition:(37)λβ1+ξX+νY+ξνXY>sX+λβν+ξνX1+ξX+νY+ξνXY2.Proof.
Let,(38)Q1X,Y=rX1−Xk−λXY1+ξX,Q2X,Y=−sY+λβXY1+ξX+νY+ξνXY.
By multiplying the Dulac functionMX,Y=1/XY by Q1X,Y and Q2X,Y, we get the following equation:(39)MQ1=1XYrX1−Xk−λXY1+ξX=rY−1kYX−λ1+ξX.
By taking the divertive ofMQ1 according to X we have the following equation:(40)∂MQ1∂X=−1kY+λξ1+ξX2.
Also forQ2X,Y we have the following equation:(41)MQ2=1XY−sY+λβXY1+ξX+νY+ξνXY=−sX+λβ1+ξX+νY+ξνXY.
By taking the divertive ofMQ2 according to Y we have the following equation:(42)∂MQ2∂Y=−λβν+ξνX1+ξX+νY+ξνXY2.
As observed, there is a change in the signs of(43)ΔMQ1,MQ2=−sX+λβ1+ξX+νY+ξνXY−λβν+ξνX1+ξX+νY+ξνXY2.
By 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:r=1.55, k=6, λ=2, ξ=1.3, β=1.2, s=0.4, x0=2.0, and y0=1.0. However, the value of the interference predator ν changes for each figure.Second case is periodic dynamics:r=2, k=4, λ=2, ξ=1.3, ν=0.9, β=1.2, s=0.4, x0=2, and y0=1. However, the value of the interference predator (ν) changes for each figure.Figures3–6 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 (Figure7), 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.Figure 3
Dynamics of the steady state case when (ν=1.25): (a) nullclines and phase portrait trajectories and (b) time series.
(a)(b)Figure 4
Dynamics of the steady state case when (ν=1.5): (a) nullclines and phase portrait trajectories and (b) time series.
(a)(b)Figure 5
Dynamics of the steady state case when (ν=2.0): (a) nullclines and phase portrait trajectories and (b) time series.
(a)(b)Figure 6
Dynamics of the steady state case when (ν=3.0): (a) nullclines and phase portrait trajectories and (b) time series.
(a)(b)Figure 7
Dynamics of the periodic case when (ν=0.9): (a) nullclines and phase portrait trajectories and (b) time series.
(a)(b)Figure 8
Dynamics of the periodic case when (ν=1.0): (a) nullclines and phase portrait trajectories and (b) time series.
(a)(b)Figure 9
Dynamics of the periodic case when (ν=1.25): (a) nullclines and phase portrait trajectories and (b) time series.
(a)(b)Figure 10
Dynamics of the periodic case when (ν=2.0): (a) nullclines and phase portrait trajectories and (b) time series.
(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 Theorem1.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 Proposition1.(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 Theorems2–4. 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.



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