Analysis of The Rosenzweig-MacArthur Predator-Prey Model with Anti-Predator Behavior


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 6(4) (2021), Pages 260-269 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: January 22, 2021 Reviewed: March 17, 2021 Accepted: April 14, 2021 
DOI: http://dx.doi.org/10.18860/ca.v6i4.11472      

Analysis of The Rosenzweig-MacArthur Predator-Prey Model with 
Anti-Predator Behavior 

Ismail Djakaria1, Muhammad Bachtiar Gaib2 , Resmawan3 

1,2,3Department of Mathematics, Universitas Negeri Gorontalo, Indonesia 

 

Email: iskar@ung.ac.id, m.tiargaib@gmail.com, resmawan@ung.ac.id 

ABSTRACT 

This paper discusses the analysis of the Rosenzweig-MacArthur predator-prey model with anti-
predator behavior. The analysis is started by determining the equilibrium points, existence, and 
conditions of the stability. Identifying the type of Hopf bifurcation by using the divergence 
criterion. It has shown that the model has three equilibrium points, i.e., the extinction of 
population equilibrium point (𝐸0), the non-predatory equilibrium point (𝐸1), and the co-existence 
equilibrium point (𝐸2). The existence and stability of each equilibrium point can be shown by 
satisfying several conditions of parameters. The divergence criterion indicates the existence of the 
supercritical Hopf-bifurcation around the equilibrium point 𝐸2. Finally, our model's dynamics 
population is confirmed by our numerical simulations by using the 4th-order Runge-Kutta 
methods. 

Keywords: Rosenzweig-MacArthur; predator-prey model; anti-predator behaviour; Hopf 
Bifurcation; divergence criterion; equilibrium point. 

INTRODUCTION 

Population dynamics are the most interesting research in mathematical biology 
which discusses the interactions that occur between prey and predator in a particular 
ecosystem [1]. This interaction has implemented to a simple mathematical model known 
as the Lotka-Volterra predator-prey model [2]. 

In a mathematical model, the predation process (interaction between prey and 
predator) is expressed in some form that is known as a functional response. This 
functional response has classified three functions, i.e. Holling-Type I, Holling-Type II, and 
Holling-Type III where each type determine the characteristic of the predator  [3]. On the 
progress, Rosenzweig and MacArthur modifying the Lotka-Volterra predator-prey model 
with the assumption the attack rate of predator increases at a decreasing rate with prey 
density until it becomes constant due to satiation which is affected by Holling-Type II 
functional response [4]. Further, some modified of Lotka-Volterra predator-prey model by 
considering the infectious disease [5]-[7]. 

Several research has discussed the modification of the Rosenzweig-MacArthur 
predator-prey model [8][9] is introduced predator foraging facilitation into Holling-Type 
II functional response. Furthermore, the Rosenzweig-MacArthur model has modified with 
various factors, e.g. the stage-structure [10][11], the refuge effect [12][13], the harvesting 
to one or more population [14][15]. From several studies described above, no one 

http://dx.doi.org/10.18860/ca.v6i4.11472
mailto:iskar@ung.ac.id
mailto:m.tiargaib@gmail.com
mailto:resmawan@ung.ac.id


 Analysis of The Rosenzweig-MacArthur Predator-Prey Model with Anti-Predator Behaviour 

Ismail Djakaria 261 

(1) 

considering anti-predator behavior factors. 
In this article, the Rosenzweig-MacArthur predator-prey model by [6] modified 

considering anti-predator behavior factors [16]. These factors can be considered in the 
model because the dynamics of the model will be very complex when the prey population 
prefers to defending and provide resistance when the predation process is occurring. The 
structure of this paper is as follows. In the next section, the methods in our work are 
described. Then, the analysis of the model has been discussed. Finally, a brief conclusion 
of our work is given. 

 

METHODS 

The dynamics of the model is analyzed by carrying out the following steps: 
1. Modifying the Rosenzweig-MacArthur predator-prey model considering anti-

predator behavior factors. 
2. Simplifying the model by using non-dimensional to reduce the number of 

parameters and solving the equilibrium points of the model. 
3. Identifying the existence, local stability, and global stability of the equilibrium 

points. 
4. Identifying the Hopf-bifurcation type by using the divergence criterion. 
5. Demonstrated the numerical simulations of the model to describe the analysis 

results by using the 4th-order Runge-Kutta method. 

RESULTS AND DISCUSSION  

Mathematical Model 

In this article, the mathematical model is formulated based on the following assumptions: 
1. The prey population is assumed to grow logistically with an intrinsic growth rate 

of 𝑟 and carrying capacity of the environment of 𝐾 and reduced due to the 
predation process. 

2. The predator population is assumed to grow due to the predation process. 𝑐 is the 
conversion rate of the consumed prey into predator births. 

3. The predation process follows Holling-Type II functional response which is 
affected by the encounter rate function where there is foraging facilitation of 
predator (𝑤 = 0), 𝑎 is the saturated rate of the predator, 𝑏 is coefficient interaction 
on both population and ℎ is the predator time handling. 

4. 𝑚 is the mortality of predators. 
5. 𝜂 is the anti-predator behavior. 
 

From the following assumptions above, the dynamics of the model can be represented by 
the following set of differential equations: 

 
𝑑𝑥

𝑑𝑡
= 𝑟𝑥(1−

𝑥

𝐾
)−

(𝑎 −𝑏)𝑥𝑦

𝑦 +ℎ(𝑎 −𝑏)𝑥
 

𝑑𝑦

𝑑𝑡
=

𝑐(𝑎 −𝑏)𝑥𝑦

𝑦 +ℎ(𝑎 −𝑏)𝑥
−𝑚𝑦 −𝜂𝑥𝑦 

 
Where 𝑥 and 𝑦 are respectively the densities of prey and predator population at time 𝑡 and 
𝑥(0),𝑦(0) > 0. 
 



 Analysis of The Rosenzweig-MacArthur Predator-Prey Model with Anti-Predator Behaviour 

Ismail Djakaria 262 

(2) 

(3) 

(4) 

To simplify our analysis, we reduce the number of parameters in system (1) by using the 
following parameter scales [17]: 

 

𝑥 → 𝑥𝐾, 𝑦 → 𝑦(𝑎 −𝑏)𝐾ℎ, 𝑡 →
𝑡

𝑟
 

 
We obtain the following non-dimensional model 
 

𝑑𝑥

𝑑𝑡
= 𝑥(1−𝑥)−

𝛼𝑥𝑦

𝑥 +𝑦
 

𝑑𝑦

𝑑𝑡
=
𝛽𝑥𝑦

𝑥 +𝑦
−𝛾𝑦 −𝛿𝑥𝑦 

 
where 

 

𝛼 =
(𝑎 −𝑏)

𝑟
, 𝛽 =

𝑐

ℎ𝑟
, 𝛾 =

𝑚

𝑟
, 𝛿 =

𝜂𝐾

𝑟
 

 

Existence and Stability Analysis of Equilibrium Points 

In this section, the equilibrium point of model (2) is obtained by solving [18]: 
 

𝑥(1−𝑥)−
𝛼𝑥𝑦

𝑥 +𝑦
= 0 

𝛽𝑥𝑦

𝑥 +𝑦
−𝛾𝑦 −𝛿𝑥𝑦 = 0 

 
Thus, from the system (3), we obtain the following equilibrium points, i.e.: 
1. A trivial equilibrium point 𝐸0 = (0,0), always exists. 
2. A non-predator equilibrium point 𝐸1 = (1,0), always exists too. 
3. A co-existence equilibrium point 𝐸2 = (𝑥

∗,𝑦∗), where 
 

𝑥∗ =
𝛽 −𝛼𝛽 +𝛼𝛾

𝛽 −𝛼𝛿
, 𝑦∗ =

(𝛽 −𝛼𝛽 +𝛼𝛾)(𝛽 −𝛾 −𝛿)

(𝛽 −𝛼𝛿)(𝛾 +𝛿 −𝛼𝛿)
 

 
which exists if 

 
𝛽 > 𝛼(𝛽 −𝛾), 𝛾 +𝛿 < 𝛼𝛿 < 𝛽 

 
Now, study the local stability of the dynamics of the system (3) around each of equilibrium 
point. The Jacobian matrix from the system (3) is determined as [19]: 
 

𝐽(𝑥,𝑦) =

(

 
1−2𝑥 −

𝛼𝑦

𝑥 +𝑦
+

𝛼𝑥𝑦

(𝑥 +𝑦)2
−
𝛼𝑥

𝑥 +𝑦
+

𝛼𝑥𝑦

(𝑥 +𝑦)2

𝛽𝑦

𝑥 +𝑦
−

𝛽𝑥𝑦

(𝑥 +𝑦)2
−𝛿𝑦

𝛽𝑥

𝑥 +𝑦
−

𝛽𝑥𝑦

(𝑥 +𝑦)2
−𝛾 −𝛿𝑥

)

  

 
By evaluating this Jacobian matrix (4) at each equilibrium point, we obtain the local 
stability properties of 𝐸0, 𝐸1, and 𝐸2 as follows. 



 Analysis of The Rosenzweig-MacArthur Predator-Prey Model with Anti-Predator Behaviour 

Ismail Djakaria 263 

 
Theorem 1. The trivial equilibrium point 𝐸0 always unstable (saddle). 
Proof:  
The Jacobian matrix (4) evaluated in equilibrium point 𝐸0 is given by 

 

𝐽(𝐸0) = (
1 0
0 −𝛾

) 

 
So, by solving the characteristic equation, we obtained the eigenvalues of 𝐽(𝐸0) is 𝜆1 = 1 

and 𝜆2 = −𝛾. It means 𝜆1 > 0 and 𝜆2 < 0. Therefore, stability of equilibrium point 𝐸0 is 
unstable (saddle).∎ 
 
Theorem 2. If 𝛿 > 𝛽 −𝛾, then the non-predatory equilibrium point 𝐸1 of system (2) is 
locally asymptotically stable. 
Proof: 
The Jacobian matrix (4) evaluated in equilibrium point 𝐸1 is given by 
 

𝐽(𝐸1) = (
−1 −𝛼
0 𝛽 −𝛾 −𝛿

) 

 
So, by solving the characteristic equation, we obtained the eigenvalues of 𝐽(𝐸1) is 𝜆1 = −1 

and 𝜆2 = 𝛽 −𝛾 −𝛿. It means 𝜆1 < 0. Therefore, if 𝛿 > 𝛽 −𝛾 then each the eigenvalues of 
𝐽(𝐸1) are negatif, and 𝐸1 is locally asymptotically stable.∎ 

 
Theorem 3. The co-existence equilibrium point 𝐸2 is locally asymptotically stable if the 
conditions below are satisfied 

𝛿2 <
Θ+Υ

Ζ
 

Proof: 
The Jacobian matrix (4) evaluated in equilibrium point 𝐸1 is given by 
 

𝐽(𝐸2) = (
𝑀11 𝑀12
𝑀21 𝑀22

) 

 
Where 

𝑀11 =
−𝛽2 +𝛼𝛽2 −𝛼𝛾2 −2𝛼𝛿(𝛼 −1)(𝛽 −𝛾)−𝛼𝛿2 +𝛼2𝛿2

(𝛽 −𝛼𝛿)2
 

𝑀12 = −
𝛼(𝛾 +𝛿 −𝛼𝛿)2

(𝛽 −𝛼𝛿)2
 

𝑀21 =
(𝛽 −𝛾 −𝛿)(𝛽2𝛾 +𝛼2𝛾𝛿2 −𝛽(𝛾2 +2𝛾𝛿 +𝛿2(𝛼 −1)2))

(𝛽 −𝛼𝛿)2
 

𝑀22 = −
𝛽(𝛽 −𝛾 −𝛿)(𝛾 +𝛿 −𝛼𝛿)

(𝛽 −𝛼𝛿)2
 

 
By solving the characteristic equation, we obtained the eigenvalues of 𝐽(𝐸2) is 

 

𝜆1,2 =
1

2
.

1

(𝛽 −𝛼𝛿)2
(𝐴±𝐵) 



 Analysis of The Rosenzweig-MacArthur Predator-Prey Model with Anti-Predator Behaviour 

Ismail Djakaria 264 

 
Where 

𝐴 = Ζ𝛿2 −Θ−Υ          and          𝐵 = Ψ2 −𝛼Ω 
With 

Ζ = (𝛼2 −𝛼 +𝛽 −𝛼𝛽) 
Θ = 𝛿(𝛽(𝛽 −2𝛾)+2𝛼2(𝛽 −𝛾)−𝛼(𝛽2 +2(𝛽 −𝛾)−𝛽𝛾)) 

Υ = 𝛽2(𝛾 −𝛼 +1)+𝛾2(𝛼 +𝛽) 

Ψ = (𝛽2 −𝛼𝛽2 +𝛼𝛾2 +2𝛼𝛿(𝛼 −1)(𝛽 −𝛾)−𝛼𝛿2(𝛼 −1)−𝛽(𝛽 −𝛾 −𝛿)(𝛾 +𝛿 −𝛼𝛿)) 

Ω = 4(𝛽 −𝛾 −𝛿)(𝛾 +𝛿 −𝛼𝛿)(𝛽2𝛾 +𝛼2𝛾𝛿2 −𝛽((𝛼 −1)2𝛿2 +𝛾2 +2𝛾𝛿)) 

 
According to (), the stability of equilibrium point 𝐸2 depending on the value of 𝐴. If 𝐴 < 0, 
we obtained: 

Ζ𝛿2 −Θ𝛿 −Υ < 0 
Ζ𝛿2 < Θ𝛿 +Υ 

𝛿2 <
Θ+Υ

Ζ
 

 
By the conditions above, the stability of equilibrium point 𝐸2 is locally asymptotically 
stable.∎ 
 
Next, study the global stability of the dynamics of the system (3) around equilibrium point 
𝐸2. We obtain the global stability properties of 𝐸2 by using the Lyapunov function [20] as 
follows. 
 
Theorem 4. The co-existence equilibrium 𝐸2 is globally asymptotically stable if the 
conditions below are satisfied: 

𝑥∗ <
(𝛼 −𝛽 +𝛾 +𝛿)(𝛾 +𝛿 −𝛼𝛿)

𝛼(𝛾 +𝛿 −𝛼𝛿)−(𝛽 −𝛾 −𝛿)2
 

 
Proof: 
Define a Lyapunov function as follows 
 

𝑉(𝑥,𝑦) = [𝑥 −𝑥∗ −𝑥∗ ln(
𝑥

𝑥∗
)]+[𝑦 −𝑦∗ −𝑦∗ ln(

𝑦

𝑦∗
)] 

 
By using the function �̇� < 0,∀ (𝑥,𝑦) ∈ ℝ2

+, we obtain: 
 

𝜕𝑉

𝜕𝑥
.
𝜕𝑥

𝜕𝑡
+
𝜕𝑉

𝜕𝑦
.
𝜕𝑦

𝜕𝑡
≤ 0 

 

(1−
𝑥∗

𝑥
)(𝑥(1−𝑥)−

𝛼𝑥𝑦

𝑥 +𝑦
)+(1−

𝑦∗

𝑦
)(
𝛽𝑥𝑦

𝑥 +𝑦
−𝛾𝑦 −𝛿𝑥𝑦) ≤ 0 

 

(
(1−𝑥)(𝑥 +𝑦)−𝛼𝑦

𝑥 +𝑦
)(𝑥 −𝑥∗)+(

𝛽𝑥 −𝛾(𝑥 +𝑦)−𝛿𝑥(𝑥 +𝑦)

𝑥 +𝑦
)(𝑦 −𝑦∗) ≤ 0 

 
For (𝑥,𝑦) ∈ ℝ2

+, we obtain: 
 



 Analysis of The Rosenzweig-MacArthur Predator-Prey Model with Anti-Predator Behaviour 

Ismail Djakaria 265 

(5) 

−𝛼 +𝛼𝑥∗ +𝛽 −𝛾 −𝛿 −(𝛽 −𝛾 −𝛿)𝑦∗ < 0 
 

−𝛼 +𝛼𝑥∗ +𝛽 −𝛾 −𝛿 −𝑥∗
(𝛽 −𝛾 −𝛿)2

(𝛾 +𝛿 −𝛼𝛿)
< 0 

 

𝑥∗ (
𝛼(𝛾 +𝛿 −𝛼𝛿)−((𝛽 −𝛾 −𝛿)2)

(𝛾 +𝛿 −𝛼𝛿)
) < 𝛼 −𝛽 +𝛾 +𝛿 

 

𝑥∗ <
(𝛾 +𝛿 −𝛼𝛿)(𝛼 −𝛽 +𝛾 +𝛿)

𝛼(𝛾 +𝛿 −𝛼𝛿)−((𝛽 −𝛾 −𝛿)2)
 

 
By the conditions above, the stability of equilibrium point 𝐸2 is globally asymptotically 
stable.∎ 
 

Analysis of Hopf Bifurcation Type 

In this section, we’ll define the Hopf-bifurcation type by using the divergence criterion 
[21]. System (3) underwent a Hopf-bifurcation when it satisfies the following conditions: 

𝛿2 <
Θ+Υ

Ζ
 and 𝛼 >

Ψ2

Ω
 

To determine the Hopf-bifurcation type of system (3) on equilibrium point 𝐸2, then we 
formed a new system. Let 𝜙(𝑥,𝑦) is a divergence of (𝑎𝑓,𝑎𝑔). We obtain the coefficient 
value of 𝑎(𝑥,𝑦) of the system (3) when the parameter value 𝛼 = 2, 𝛽 = 0.79, 𝛾 = 0.5, and 
𝛿 = 0.0186 with equilibrium point 𝐸2

∗ = (0.279;0.157) as follows: 
 

𝑎(𝑥,𝑦) = 1+6.956𝑥 +13,386𝑦 −6.77𝑥2 +32.968𝑥𝑦 +55.507𝑦2 
 

So that a new system is obtained: 
𝑧(𝑥,𝑦) = (1+6.956𝑥 +13,386𝑦 −6.77𝑥2 +32.968𝑥𝑦 +55.507𝑦2) 

(𝑥(1−𝑥)−
𝛼𝑥𝑦

𝑥 +𝑦
) 

𝑤(𝑥,𝑦) = (1+6.956𝑥 +13,386𝑦 −6.77𝑥2 +32.968𝑥𝑦 +55.507𝑦2) 

(
𝛽𝑥𝑦

𝑥 +𝑦
−𝛾𝑦 −𝛿𝑥𝑦) 

By linearizing system (4), we obtained: 
 

𝐽(𝐸2
∗) = (

1.337 −6.002
0.732 −1.337

) 

 
By solving the characteristic equation, we obtained the eigenvalues of 𝐽(𝐸2

∗) is 

 
𝜆1,2 = ±1.615𝑖 

 
For a system (5) to obtain the eigenvalues of conjugate complex numbers, then we can 

analyze the Hopf-bifurcation of system (3) type by looking at the divergence value of 
system (3). We obtained: 

 
𝜙𝑥𝑥(𝐸2

∗) = −21.109 
 



 Analysis of The Rosenzweig-MacArthur Predator-Prey Model with Anti-Predator Behaviour 

Ismail Djakaria 266 

Based on the divergence value above, a stable limit cycle appears in the system (3). 
Therefore, system (3) underwent a Supercritical Hopf-bifurcation. 
 

Numerical Simulations 

In this section, the numerical simulation is solved using the 4th-order Runge-Kutta 
method [22] with initial conditions and some values of the parameters. We choose the 
following set of parameter values: 

𝛼 = 2, 𝛽 = 0.79, 𝛾 = 0.5 
With different parameter control values as follows 𝛿1 = 0.011, 𝛿2 = 0.0186 and 𝛿3 =

0.026. We using the initial condition is 𝑥(0) = 0.3 and 𝑦(0) = 0.3. 
 

 
(a) (b) 

 
Figure 1. (a) Phase Portrait of Case 1 and (b) Time-Series Portrait 

 

 
(a) (b) 

 
Figure 2. (a) Phase Portrait of Case 2 and (b) Time-Series Portrait 

 
In case 1, we obtained the dynamics of the solution on the system (3) with parameter 

control values 𝛿1 = 0.011. Based on figure 
1(a), the trivial equilibrium point 𝐸0 = (0,0) is unstable (saddle)  with eigenvalues 

𝜆1 = 1 and 𝜆2 = −0.5. This coincides with Theorem 1. The non-predator equilibrium 
point 𝐸1 = (1,0) is unstable (saddle) with eigenvalues 𝜆1 = −1 and 𝜆2 = 0.279.  This 
coincides with Theorem 2 on condition 𝛿 < 𝛽 −𝛾. The co-existence equilibrium point 
𝐸2 = (0.273;0.156) is unstable (spiral) with eigenvalues 𝜆1,2 = 0.003±0.220𝑖. This 



 Analysis of The Rosenzweig-MacArthur Predator-Prey Model with Anti-Predator Behaviour 

Ismail Djakaria 267 

coincides with Theorem 3 on condition 𝛿2 <
Θ+Υ

Ζ
. Based on figure 1(b), the prey 

population and predator population have increased and decreased of total populations. 
The case continuously oscillates with a greater deviation value. Hence, both population is 
unstable to a specific point. 
 

 
(a) (b) 

 
Figure 3. (a) Phase Portrait of Case 3 and (b) Time-Series Portrait 

 
In case 2, we obtained the dynamics of the solution on the system (3) with parameter 

control values 𝛿1 = 0.0186. Based on figure 2(a), the trivial equilibrium point 𝐸0 = (0,0) 
is unstable (saddle)  with eigenvalues 𝜆1 = 1 and 𝜆2 = −0.5. This coincides with Theorem 
1. The non-predator equilibrium point 𝐸1 = (1,0) is unstable (saddle) with eigenvalues 
𝜆1 = −1 and 𝜆2 = 0.271. This coincides with Theorem 2 on condition 𝛿 < 𝛽 −𝛾. The co-
existence equilibrium point 𝐸2 = (0.279;0.157) is center (spiral) with eigenvalues 𝜆1,2 =

±0.220𝑖. This coincides with Theorem 3 on condition 𝛿2 =
Θ+Υ

Ζ
. Based on figure 2(b), 

the oscillations that occur have a smaller deviation value. This condition explains that 
there is a stability transition from unstable to stable to a specific point. This stability 
transition has led to the appearance of Hopf-bifurcation. 

In case 3, we obtained the dynamics of the solution on the system (3) with parameter 
control values 𝛿1 = 0.026. Based on figure 3(a), the trivial equilibrium point 𝐸0 = (0,0) 
is unstable (saddle)  with eigenvalues 𝜆1 = 1 and 𝜆2 = −0.5. This coincides with Theorem 
1. The non-predator equilibrium point 𝐸1 = (1,0) is unstable (saddle) with eigenvalues 
𝜆1 = −1 and 𝜆2 = 0.264. This coincides with Theorem 2 on condition 𝛿 < 𝛽 −𝛾. The co-
existence equilibrium point 𝐸2 = (0.285;0.159) is stable (spiral) with eigenvalues 𝜆1,2 =

−0.003±0.220𝑖. This coincides with Theorem 3 on condition 𝛿2 >
Θ+Υ

Ζ
. Based on figure 

3(b), the dynamics between prey and predator begin to stabilize at 1500 days to a specific 
point. 

CONCLUSIONS 

The Rosenzweig-MacArthur predator-prey model with anti-predator behavior has 
been studied. From the analysis of system (2), we obtain three equilibrium points, i.e., the 
trivial equilibrium point (𝐸0), the non-predatory equilibrium point (𝐸1), and the co-
existence equilibrium point (𝐸2). The local stability conditions of each equilibrium point 
have been appointed, and the global stability conditions of the co-existence equilibrium 



 Analysis of The Rosenzweig-MacArthur Predator-Prey Model with Anti-Predator Behaviour 

Ismail Djakaria 268 

point (𝐸2) have been obtained. Our analysis also showed that the model occurs a 
Supercritical Hopf-bifurcation by using the divergence criterion. Numerical analytic has 
been simulated to verify the theoretical results. No one extinction matters in any 
population. 

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