Aswar Anas and Marsidi 

Michaelis-Menten Models with Constant Harvesting of Restricted Prey Populations  

Minimum Place and Amount Capacity 

CONTACT:  

Marsidi,          marsidiarin@gmail.com           Departement of  Mathematics Education Universitas PGRI 

Argopuro Jember, Jember, Jawa Timur 68121, Indonesia 

 

       The article can be accessed here. https://doi.org/10.15642/mantik.2021.7.2.107-114  

Michaelis-Menten Models with Constant Harvesting of Restricted 

Prey Populations Minimum Place and Amount Capacity 
 

 

 Aswar Anas1, Marsidi2*  

1,2Department of Mathematics Education, Universitas PGRI Argopuro, Jember, Indonesia 
 

 

Article history: 

Received Jun 16, 2021 

Revised, Jul 14, 2021 

Accepted, Oct 31, 2021 

 

Kata Kunci:  

Model rantai 

makanan, Model 

Michaelis-Menten, 

Model mangsa dan 

pemangsa 

 

Abstrak. Pemodelan rantai makanan saat ini sedang berkembang 

pesat. Ekosistem terlindungi dari rantai proses makan dan 

memakan. Semua makhluk hidup saling membutuhkan, namun jika 

proses memakannya tidak seimbang, maka kepunahan makhluk 

hidup akan terjadi. Salah satunya adalah model mangsa dan 

pemangsa yang berfungsi sebagai penyeimbang dalam sistem rantai 

makanan. Model Michaelis-Menten merupakan model mangsa-

pemangsa yang pada intinya menjaga kepunahan mangsa. 

Permasalahannya adalah bagaimana menjaga mangsa tidak punah 

namun dengan pemanenan maksimal di suatu tempat dan jumlah 

minimum mangsa dengan waktu yang tepat. Metode yang 

digunakan untuk mengatasi masalah tersebut adalah menambah dua 

variabel baru pada model Michaelis-Menten, yaitu jumlah minimum 

mangsa dan kapasitas tempat yang akan ditempati. Terlihat bahwa 

sistem akan berada dalam keseimbangan jika tingkat kematian 

pemangsa besar, sehingga mangsa terjaga dari kepunahan sampai 

pemanenan. Selain itu waktu yang tepat untuk perkembangbiakan 

yang baik juga dapat ditentukan. Dari model ini didapatkan waktu 

yang tepat untuk pemanenan agar tidak terjadi kepunahan mangsa 

adalah ℎ = 𝑛(
𝐾2−𝑚2

4 𝐾
).   

 

Keywords: 

Food chain model, Mich-

aelis-Menten model, 

Prey-predator model    

 

 

Abstract. Food chain modeling is currently developing rapidly. The 

ecosystem is protected from the chain of eating and eating pro-

cesses. All living things need each other, but if the process of eating 

them is not balanced, then the extinction of living things will occur. 

One of them is the prey and predator model that serves as a balancer 

in the food chain system. The Michaelis-Menten model is a prey-

predator model that essentially prevents prey extinction. The prob-

lem is how to keep the prey from becoming extinct but with maxi-

mum harvesting in one place and the minimum amount of prey at 

the right time. The method used to overcome this problem is to add 

two new variables to the Michaelis-Menten model, namely the min-

imum number of prey and the capacity of the place to be occupied. 

It is seen that the system will be in equilibrium if the predator mor-

tality rate is large so that the prey is kept from extinction until har-

vesting. In addition, the right time for good breeding can also be 

determined. From this model, it is found that the right time for har-

vesting so that prey extinction does not occur is ℎ = 𝑛(
𝐾2−𝑚2

4 𝐾
). 

 

How to cite: 

A. Anas and Marsidi, “Michaelis-Menten Models with Constant Harvesting of Restricted Prey 

Populations Minimum Place and Amount Capacity”, J. Mat. Mantik, vol. 7, no. 2, pp. 107-114, 

October 2021. 

  

Jurnal Matematika MANTIK 

Vol. 7, No. 2, October 2021, pp. 107-114 

 

ISSN: 2527-3159 (print) 2527-3167 (online) 

mailto:marsidiarin@gmail.com
http://u.lipi.go.id/1458103791


 

 

 
Jurnal Matematika MANTIK  

Vol 7, No 2, October 2021, pp. 107-114 
 

 

 

108 

 

1. Introduction 

The food chain is a system created in nature. The dependence of living things on other 

living things is a normal interaction on this earth. This interaction occurs because of the 

needs of one party or two parties. The relationship between prey and predators is a one-

sided relationship that harms one party, this is very closely related because predators can 

only survive if there is prey. As a result, the chances of predators experiencing extinction 

are small. In addition, the predator also functions as a controller of the growth rate of the 

prey [1]. 

Dongmei et al. stated that population extinction occurred because the initial population 

was too low [2]. If this happens, the prey population will be decrease and extinction may 

occur. As a result, predator populations are increasingly threatened indirectly. With the 

extinction of the Prey, the Predator also experiences extinction. The prey in this model is 

harvested while the predator population is not harvested because it has no commercial 

value. They are states that the commercial value is determined by the maximum value of h 

because if the prey harvested exceeds the value of h, the prey will become extinct. 

The cause of the extinction of the Prey is also influenced by over-harvesting. There-

fore, it is necessary to limit the number of prey so that the number of prey remains under 

control and does not exceed the existing capacity [3] .  

The Lotka–Volterra model is frequently used to describe the dynamics of ecological 

systems in which two species interact, one a predator and one its prey. The model is sim-

plified with the following assumptions: (1) only two species exist: fox and rabbit; (2) rab-

bits are born and then die through predation or inherent death; (3) foxes are born and their 

birth rate is positively affected by the rate of predation, and they die naturally [4]. The 

Michaelis-Menten model is a predator-prey model which is a generalization of the Lotka-

Volterra model [5]. This model is used for events if the interaction system between indi-

viduals in a population has limited capacity. However, in this case, harvesting is also ap-

plied. Previously, researchers researched the Lotka-Volterra Proportional Predation and 

Prey model on the Von Bertallanfy [6] logistics modifications model . In this model, the 

researcher concludes that the number of predators in a place is highly dependent on the 

initial number of prey, the birth rate of the predator, and the birth rate of the prey. If the 

initial number of prey is large and the birth rate of prey is also large, the growth rate of 

predators increases. And if the birth constant of predators is small and the number of pred-

ators is smaller than the prey, there will be a decrease in the growth rate of prey. From 

previous research, researchers are interested in researching the Michaelis-Menten model, 

because this model prevents the extinction of both prey and predators. However, the re-

searchers added the variables of space capacity and the minimum amount of prey that must 

be given. 

 

2. Research Method 

This research method is a literature study of the addition of the [7] variable with the 

capacity of the place and the minimum number of prey, the equation comes from: 

According to [7], the equation for population growth and space capacity is formed by 

the formula: 
𝑑𝑃(𝑡)

𝑑𝑡
= 𝑛𝑃(𝑡)(1−

𝑃(𝑡)

𝐾
)  (1) 

 

From [8] with the addition of the assumption that the minimum number of population also 

affects the rate of population growth, then the form is: 

 
𝑑𝑃(𝑡)

𝑑𝑡
= 𝑛𝑃(𝑡)(1−

𝑃(𝑡)

𝐾
)(1−

𝑚

𝑃(𝑡)
) (2) 

  



 
 

 

Aswar Anas and Marsidi 

Michaelis-Menten Models with Constant Harvesting of Restricted Prey Populations  

Minimum Place and Amount Capacity 

109 

 

where n is the coefficient of the growth rate of the prey population, P(t) is the number of 

prey populations at t, K is the capacity of the place, and m is the minimum number of 

populations. 

Equation (2) is a logistic model that is influenced by the capacity of the place and the 

minimum number population. If the growth rate is also affected by the number of harvests 

of h, then equation (2) changes to: 

 
𝑑𝑃(𝑡)

𝑑𝑡
= 𝑛𝑃(𝑡)(1−

𝑃(𝑡)

𝐾
)(1−

𝑚

𝑃(𝑡)
)−ℎ (3) 

 

Where 0 ≤ ℎ ≤ ℎ𝑚𝑎𝑥, h is the prey harvest rate constant and is the maximum harvested 
prey. 

Xiao, et al describe the general formula for the persistence model in prey as follows 

[9]: 
𝑑𝑃(𝑡)

𝑑𝑡
= 𝑛𝑃(𝑡)(1−

𝑃(𝑡)

𝐾
)−

𝑐𝑃(𝑡)𝑦(𝑡)

𝑟𝑦(𝑡)+𝑃(𝑡)
−ℎ  

𝑑𝑦(𝑡)

𝑑𝑡
= 𝑦(𝑡)(−𝐷 +

𝑓𝑃(𝑡)

𝑟𝑦(𝑡)+𝑃(𝑡)
) (4) 

 

where 𝑟 is the satisfaction level of the Predator, 𝑐 the number of prey captured, f the 
cconversion factor between the number of predators born for each prey captured, 𝐷 is the 
mortality rate of the predator. 

By adding the factor of the minimum number of prey population  to reproduce, it is 

obtained  
𝑑𝑃(𝑡)

𝑑𝑡
= 𝑛𝑃(𝑡)(1−

𝑃(𝑡)

𝐾
)(1−

𝑚

𝑃(𝑡)
)−

𝑐𝑃(𝑡)𝑦(𝑡)

𝑟𝑦(𝑡)+𝑃(𝑡)
−ℎ  

𝑑𝑦(𝑡)

𝑑𝑡
= 𝑦(𝑡)(−𝐷 +

𝑓𝑃(𝑡)

𝑟𝑦(𝑡)+𝑃(𝑡)
) (5) 

 

By [10] Equation (5) is sought a solution so that the prey does not experience extinction 

with: 

a. Look for the maximum harvest value if predator is not present. 

b. Search for fixed points. 

c. Analysis of system stability at points 𝑇1 and 𝑇2. 
 

3. Discussion  
 

The following will discuss certain limits of harvesting a population to prevent extinc-

tion, then look for a fixed point from the Michaelis-Menten model to analyze the stability 

of the system at each of these fixed points and also perform simulations with different 

parameters. 

 

3.1. Finding of Harvesting Maximum Value (𝒉𝒎𝒂𝒙) Without Predator Condition  
 

In equation (2) we have maximum points if: 

 
𝑑𝑃(𝑡)

𝑑𝑡
= 𝑛𝑃(𝑡)(1−

𝑃(𝑡)

𝐾
)(1−

𝑚

𝑃(𝑡)
) = 0 (6) 

 

By [11] with a condition half of the carrying capacity and the minimum number of prey 

population 𝑃(𝑡). In this case, fixed point occurs in 𝑃(𝑡) = 𝑚 and 𝑃(𝑡) = 𝐾. If 𝑃(𝑡) = 𝑚 
then this point is not stable. It is caused by the population 𝑃(𝑡) > 𝑚. It means that the 
population will grow rapidly and avoid the 𝑃(𝑡) = 𝑚 and will attain 𝑃(𝑡) = 𝐾. Let 𝑚 ≤

𝑃0 < (
𝐾+𝑚

2
), point 𝑃0  moves rapidly towards the maximum point when 𝑃0 = (

𝐾+𝑚

2
). 



 

 

 
Jurnal Matematika MANTIK  

Vol 7, No 2, October 2021, pp. 107-114 
 

 

 

110 

 

Whereas if (
𝐾+𝑚

2
) < 𝑃0 ≤ 𝐾, then this point moves slowly towards the stable point by K. 

If 0 ≤ 𝑃0 < 𝑚, then the population will be moving down so that the population experiences 
extinction. While if 𝐾 < 𝑃0 ,  then this point will be moving slowly down to stable point 
K.  

Since harvesting results must be maximum, then the predator must be eliminated or 

equal to zero and 𝑃(𝑡) =
𝐾+𝑚

2
, such that we have 

   
𝑑𝑃(𝑡)

𝑑𝑡
= 𝑛𝑃(𝑡)(1−

𝑃(𝑡)

𝐾
)(1−

𝑚

𝑃(𝑡)
)−ℎ = 0  

⟹ 𝑛𝑃(𝑡)(1−
𝑃(𝑡)

𝐾
)(1−

𝑚

𝑃(𝑡)
) = ℎ  

⟹
𝑛(𝐾−𝑃(𝑡))(𝑃(𝑡)−𝑚)

𝐾
= ℎ ; 𝑃(𝑡) =

𝐾+𝑚

2
 

 

So that: 

 

𝑛(
𝐾+𝑚

2
)(1−

𝐾+𝑚

2𝐾
)(1−

2𝑚

(𝐾+𝑚)
) = ℎ  

 

⟹ 𝑛(
𝐾+𝑚

2
)(

2𝐾−𝐾+𝑚

2𝐾
)(

𝐾+𝑚−2𝑚

𝐾+𝑚
) = ℎ  

 

⟹
1

2
𝑛(

𝐾+𝑚

2𝐾
)(𝐾 −𝑚) = ℎ  

 

⟹ 𝑛(
𝐾2−𝑚2

4𝐾
) = ℎ  

Therefore, the maximum harvesting is bounded by ℎ = 𝑛(
𝐾2−𝑚2

4𝐾
). 

 

3.2. Finding Fixed Point 

 

Let 

𝑓1(𝑃,𝑦) = 𝑛𝑝(1−
𝑃

𝐾
)(1−

𝑚

𝑃
)−

𝑐𝑃𝑦

𝑟𝑦+𝑃
−ℎ  

𝑓2(𝑃,𝑦) = 𝑦(−𝐷 +
𝑓𝑝

𝑟𝑦+𝑃
) (7) 

 

Suppose  𝑓1(𝑃,𝑦) = 0;  𝑓2(𝑃,𝑦) = 0 such that we have 𝑦 = 0 or (−𝐷 +
𝑓𝑝

𝑟𝑦+𝑃
) = 0. 

 

(−𝐷 +
𝑓𝑝

𝑟𝑦+𝑃
) = 0  

⟺
𝑓𝑝

𝑟𝑦+𝑃
= 𝐷  

⟺ 𝑓𝑃 = 𝐷𝑟𝑦 +𝑃𝐷  

⟺
𝑓𝑃−𝑃𝐷

𝐷𝑟
= 𝑦  

 

Then we have  𝑦1 = 0; 𝑦2 =
𝑃(𝑓−𝐷)

𝐷𝑟
. Furthermore, finding the points 𝑃1  and 𝑃2  by 

substituting 𝑦1 = 0 to 𝑓1(𝑃,𝑦1) = 0. 
 

𝑓1(𝑃,𝑦) = 𝑛𝑝(1−
𝑃

𝐾
)(1−

𝑚

𝑃
)−

𝑐𝑃0

𝑟0+𝑃
−ℎ = 0  

⟺ 𝑛𝑝(1−
𝑃

𝐾
)(1−

𝑚

𝑃
)−ℎ = 0  



 
 

 

Aswar Anas and Marsidi 

Michaelis-Menten Models with Constant Harvesting of Restricted Prey Populations  

Minimum Place and Amount Capacity 

111 

 

⟺ 𝑛𝑝(
𝐾−𝑃

𝐾
)(

𝑝−𝑚

𝑃
)−ℎ = 0  

⟺
𝑛

𝐾
(𝐾𝑃 −𝐾𝑚 +𝑃𝑚−𝑃2)−ℎ = 0  

⟺ 𝑛𝑃 −𝑛𝑚 +
𝑃

𝐾
𝑚𝑛 −

𝑛

𝐾
𝑃2 −ℎ = 0  

⇔ −
𝑛

𝐾
𝑃2 +𝑃(𝑛 +

𝑚𝑛

𝐾
)−(ℎ +𝑛𝑚) = 0  

⇔
𝑛

𝐾
𝑃2 −(𝑛 +

𝑚𝑛

𝐾
)𝑃 +(ℎ +𝑛𝑚) = 0  

Such that 

 

 𝑃1,2 =
(𝑛+

𝑚𝑛

𝐾
)±√(−(𝑛+

𝑚𝑛

𝐾
))
2

−4(
𝑛

𝐾
)(ℎ+𝑛𝑚)

2(
𝑛

𝐾
)

 

𝑃1,2 =
(𝑛+

𝑚𝑛

𝐾
)±√(𝑛+

𝑚𝑛

𝐾
)
2
−4(

𝑛

𝐾
)(ℎ+𝑛𝑚)

2(
𝑛

𝐾
)

  

 

For 𝑦2 =
𝑃(𝑓−𝐷)

𝐷𝑟
  we have 

 

𝑛𝑝(1−
𝑃

𝐾
)(1−

𝑚

𝑃
)−(

𝑐𝑃
𝑃(𝑓−𝐷)

𝐷𝑟

𝑟
𝑃(𝑓−𝐷)

𝐷𝑟
 +𝑃
)−ℎ = 0  

⇔ 𝑛𝑝(1 −
𝑃

𝐾
)(1−

𝑚

𝑃
)−(

𝑐𝑃
𝑃(𝑓−𝐷)

𝐷𝑟

𝑟
𝑃(𝑓−𝐷)+𝑃𝐷𝑟

𝐷𝑟
 
)−ℎ = 0  

⇔ 𝑛𝑝(1 −
𝑃

𝐾
)(1−

𝑚

𝑃
)−

(𝑐𝑃2𝑓−𝑐𝑃2𝐷)

𝑟𝑃𝑓−𝑃𝐷𝑟+𝑃𝐷𝑟
−ℎ = 0  

⇔ 𝑛𝑝(1 −
𝑃

𝐾
)(1−

𝑚

𝑃
)−

(𝑐𝑃2𝑓−𝑐𝑃2𝐷)

𝑟𝑃𝑓
−ℎ = 0  

⇔
𝑛

𝐾
(𝐾𝑃 −𝐾𝑚 +𝑃𝑚−𝑃2)−

(𝑐𝑃2𝑓−𝑐𝑃2𝐷)

𝑟𝑃𝑓
−ℎ = 0  

⇔
𝑟𝑃𝑓𝑛

𝐾
(𝐾𝑃 −𝐾𝑚 +𝑃𝑚 −𝑃2)−(𝑐𝑃2𝑓 −𝑐𝑃2𝐷)−ℎ𝑟𝑃𝑓 = 0  

⇔ 𝑃(
𝑟𝑓𝑛

𝐾
(𝐾𝑃 −𝐾𝑚 +𝑃𝑚−𝑃2)−(𝑐𝑃𝑓 −𝑐𝑃𝐷)−ℎ𝑟𝑓) = 0  

⇔ 𝑃((𝑟𝑓𝑛𝑃 −𝑟𝑓𝑛𝑚 +
𝑟𝑓𝑛𝑃𝑚

𝐾
−
𝑟𝑓𝑛𝑃2

𝐾
)−𝑐𝑃𝑓 +𝑐𝑃𝐷 = ℎ𝑟𝑓) = 0  

⇔ 𝑃(−
𝑟𝑓𝑛

𝐾
𝑃2 +𝑃(𝑟𝑓𝑛+

𝑟𝑓𝑛𝑚

𝐾
−𝑐𝑓 +𝑐𝐷)−(ℎ𝑟𝑓 +𝑟𝑛𝑓𝑚)) = 0  

 

⇔ 𝑃(
𝑟𝑓𝑛

𝐾
𝑃2 −(𝑟𝑓𝑛+

𝑟𝑓𝑛𝑚

𝐾
−𝑐𝑓 +𝑐𝐷)𝑃 +(ℎ𝑟𝑓 +𝑟𝑛𝑓𝑚)) = 0  

 

Thus 𝑃 = 0 or  

𝑃1,2 =
(𝑟𝑓𝑛+

𝑟𝑓𝑛𝑚

𝐾
−𝑐𝑓+𝑐𝐷)±√(−(𝑟𝑓𝑛+

𝑟𝑓𝑛𝑚

𝐾
−𝑐𝑓+𝑐𝐷))

2

−4(
𝑟𝑓𝑛

𝐾
)(ℎ𝑟𝑓+𝑟𝑛𝑓𝑚)

2(
𝑟𝑓𝑛

𝐾
)

  

𝑃1,2 =
(𝑟𝑓𝑛

(𝐾+𝑚)

𝐾
−𝑐𝑓+𝑐𝐷)±√(𝑟𝑓𝑛+

𝑟𝑓𝑛𝑚

𝐾
−𝑐𝑓+𝑐𝐷)

2
−4(

𝑟𝑓𝑛

𝐾
)(ℎ𝑟𝑓+𝑟𝑛𝑓𝑚)

2(
𝑟𝑓𝑛

𝐾
)

  

 

It is impossible for 𝑃 = 0. Since this model must have a minimum amount so that the prey 
can grow. 

From the results above, we have a fixed points as follows: 



 

 

 
Jurnal Matematika MANTIK  

Vol 7, No 2, October 2021, pp. 107-114 
 

 

 

112 

 

 

𝑇1:(𝑃1,𝑦1) =

(

 
 
(𝑛+

𝑚𝑛

𝐾
)+√((𝑛+

𝑚𝑛

𝐾
))
2

−4(
𝑛

𝐾
)(ℎ+𝑛𝑚)

2(
𝑛

𝐾
)

,0

)

 
 

  

𝑇2:(𝑃2,𝑦1) = (
(𝑛+

𝑚𝑛

𝐾
)−√(𝑛+

𝑚𝑛

𝐾
)
2
−4(

𝑛

𝐾
)(ℎ+𝑛𝑚)

2(
𝑛

𝐾
)

,0)  

𝑇3:(𝑃1
∗,𝑦2

∗) = (
(𝑟𝑓𝑛

(𝐾+𝑚)

𝐾
−𝑐𝑓+𝑐𝐷)+√(𝑟𝑓𝑛+

𝑟𝑓𝑛𝑚

𝐾
−𝑐𝑓+𝑐𝐷)

2
−4(

𝑟𝑓𝑛

𝐾
)(ℎ𝑟𝑓+𝑟𝑛𝑓𝑚)

2(
𝑟𝑓𝑛

𝐾
)

,
𝑃1
∗(𝑓−𝐷)

𝐷𝑟
)  

𝑇4:(𝑃2
∗,𝑦2

∗) = (
(𝑟𝑓𝑛

(𝐾+𝑚)

𝐾
−𝑐𝑓+𝑐𝐷)−√(𝑟𝑓𝑛+

𝑟𝑓𝑛𝑚

𝐾
−𝑐𝑓+𝑐𝐷)

2
−4(

𝑟𝑓𝑛

𝐾
)(ℎ𝑟𝑓+𝑟𝑛𝑓𝑚)

2(
𝑟𝑓𝑛

𝐾
)

,
𝑃2
∗(𝑓−𝐷)

𝐷𝑟
)  

 

Equilibrium analysis is carried out to find out the points that cause the system to be in 

equilibrium and not, to analyze it, look for real eigenvalues at each equilibrium point [12]. 

To find the stability of the equilibrium point of a model, you can use the Jacobian matrix 

with the order  2×2. This matrix is discussed in the next sub-chapter.  
 

3.3. Jacobian Matrix 
 

In [13] let the equation system equation (5) written by: 

 
𝑑𝑃

𝑑𝑡
= 𝑓1(𝑃,𝑦)  

𝑑𝑌

𝑑𝑡
= 𝑓2(𝑃,𝑦)  

 

Such that, the Jacobian matrix can be formed by: 

 

𝐽 = [

𝜕𝑓1

𝜕𝑃

𝜕𝑓1

𝜕𝑦

𝜕𝑓2

𝑃

𝜕𝑓2

𝜕𝑦

]  

 

𝐽 = [
−
2𝑛𝑝

𝐾
+
𝑛(𝐾+𝑚)

𝐾
−
𝑟𝑐𝑦2+𝑐𝑦𝑃+𝑟𝑐𝑃𝑦2

(𝑟𝑦+𝑃)2
−

𝑐𝑃2

(𝑟𝑦+𝑃)2

𝑓𝑟𝑦2

(𝑟𝑦+𝑃)2
−𝐷 +

𝑓𝑟𝑦2+𝑓𝑦𝑃−𝑓𝑃𝑦𝑟

(𝑟𝑦+𝑃)2

]  

 

The stability of the system of equations (5) will be known by analyzing the eigen values of 

the Jacobian matrix. 

 

3.4. Fixed Point Stable Analysis 
 

The following discussion discusses the stability analysis of fixed points using the Ja-

cobi matrix of order 2×2 and only discussed in point 𝑇1(𝑃1,𝑦1), 𝑇2(𝑃2,𝑦1). 
 

3.4.1 Stable system at fixed point 𝑻𝟏 

By [14] we take the fixed point 

 



 
 

 

Aswar Anas and Marsidi 

Michaelis-Menten Models with Constant Harvesting of Restricted Prey Populations  

Minimum Place and Amount Capacity 

113 

 

𝑇1:(𝑃1,𝑦1) = (
(𝑛+

𝑚𝑛

𝐾
)+√(𝑛+

𝑚𝑛

𝐾
)
2
−4(

𝑛

𝐾
)(ℎ+𝑛𝑚)

2(
𝑛

𝐾
)

,0). We substitute the point 𝑇1 to Jacobian 

Matrix as follows. 

 

𝐽1 = [
−
2𝑛𝑝

𝐾
+
𝑛(𝐾+𝑚)

𝐾
−
𝑟𝑐𝑦2+𝑐𝑦𝑃+𝑟𝑐𝑃𝑦2

(𝑟𝑦+𝑃)2
−

𝑐𝑃2

(𝑟𝑦+𝑃)2

𝑓𝑟𝑦2

(𝑟𝑦+𝑃)2
−𝐷 +

𝑓𝑟𝑦2+𝑓𝑦𝑃−𝑓𝑃𝑦𝑟

(𝑟𝑦+𝑃)2

]   

 

Then we have: 

 

𝐽1 = [
−
2𝑛𝑝

𝐾
+
𝑛(𝐾+𝑚)

𝐾
−
𝑟𝑐0+𝑐0𝑃+𝑟𝑐𝑃0

(𝑟0+𝑃)2
−

𝑐𝑃2

(𝑟0+𝑃)2

0 −𝐷
]  

 

𝐽1 = [
−
2𝑛𝑝

𝐾
+
𝑛(𝐾+𝑚)

𝐾
−𝑐

0 −𝐷
]  

 

𝐽1 = [
−𝑛 −

𝑚𝑛

𝐾
−√(𝑛 +

𝑚𝑛

𝐾
)−

4𝑛(𝑚𝑛+ℎ)

𝐾
+
𝑛(𝐾+𝑚)

𝑚
−𝑐

0 −𝐷

]  

 

𝐽1 = [
−√(𝑛 +

𝑚𝑛

𝐾
)
2
−
4𝑛(𝑚𝑛+ℎ)

𝐾
−𝑐

0 −𝐷

]  

 

The eigen value can be obtained by 𝑑𝑒𝑡(𝐴−𝜆1𝐼) = 0. 

(−√(𝑛+
𝑚𝑛

𝐾
)
2
−
4𝑛(𝑚𝑛+ℎ)

𝐾
−𝜆)(−𝐷 −𝜆) = 0  

⇔ 𝜆1 = −√(𝑛 +
𝑚𝑛

𝐾
)
2
−
4𝑛(𝑚𝑛+ℎ)

𝐾
   or  𝜆2 = −𝐷  

 

with ℎ = 𝑛(
𝐾2−𝑚2

4𝐾
) we have 𝜆1 = −√2√−

𝑛2𝑚(𝐾−𝑚)

𝐾2
   or  𝜆2 = −𝐷. 

 

From the equation above we know that  𝜆1 is not real eigen value because 𝑛,𝑚,𝐾 is a pos-
itive parameter with 𝐾 > 𝑚 and 𝜆2 ≤ 0 if 𝐷 is a positive integer. It show that 𝑇1 is stable.  
 

3.4.2 Stable system at fixed point 𝑻𝟐 

Let  𝑇2:(𝑃2,𝑦1) = (
(𝑛+

𝑚𝑛

𝐾
)−√(𝑛+

𝑚𝑛

𝐾
)
2
−4(

𝑛

𝐾
)(ℎ+𝑛𝑚)

2(
𝑛

𝐾
)

,0). With the same way we obtain that 

𝜆1 = √(𝑛 +
𝑚𝑛

𝐾
)
2
−
4𝑛(𝑚𝑛+ℎ)

𝐾
   or 𝜆2 = −𝐷. Wit ℎ = 𝑛(

𝐾2−𝑚2

4𝐾
) we get                          𝜆1 =

√2√−
𝑛2𝑚(𝐾−𝑚)

𝐾2
   or  𝜆2 = −𝐷, because 𝜆1 is not real eigen value and 𝜆2 ≤ 0 if 𝐷 is a 

positive integer. It show that 𝑇2 is stable  [15].  
 

  



 

 

 
Jurnal Matematika MANTIK  

Vol 7, No 2, October 2021, pp. 107-114 
 

 

 

114 

 

4. Conclusion 
 

From the calculation results obtained 4 fixed points. In here we are only discussing 

points that are 𝑇1and 𝑇2. The analysis carried out at two fixed points states that the equilib-
rium and stability of the population is influenced by the level of capacity, the minimum 

number of population, and the death rate of the predator. From the discussion section we 

know that maximum harvesting can be done if ℎ = 𝑛(
𝐾2−𝑚2

4𝐾
), it’s aim is to ensure that 

prey and predator populations do not become extinct. 
 

5. Acknowledgments 

We gratefully acknowledge the support from Universitas PGRI Argopuro Jember 

2021. 

 

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