Lotka-Volterra Model with Periodic Harvesting
NORMA MUHTAR1,2, EDI CAHYONO1, R. MARSUKI ISWANDI2, MUHIDIN2
1Department of Mathematics, FMIPA, Universitas Halu Oleo, INDONESIA
2Department of Agriculture, Graduate Program, Universitas Halu Oleo, INDONESIA
Abstract. A closed interaction of predator prey is considered. The interaction is expressed in the Lotka-Volterra
model. Two types of Lotka-Volterra models are considered, with and without carrying capacity of the prey. The
paper includes a periodic harvesting of predator and/or prey, a function of time which acts to the model. Hence,
the model is in the form of a system of non-homogeneous equations. Dynamical properties of the models are
investigated. The solutions are computed numerically. Such interaction is in the need of integrated farming on
harvesting of predator and/or prey. In this model the number of population in the system is sensitive to the
initial value, which can be applied to the integrated farming systems such that the system remains sustainable.
Key-Words: Dynamical properties, integrated farming, Lotka-Volterra model, prey-predator.
Received: March 11, 2022. Revised: November 2, 2022. Accepted: December 5, 2022. Published: December 31, 2022.
1 Introduction
After the model was introduced by Lotka, [1], and
Volterra [2], it attracted many researchers which
resulted in many papers. Although Lotka used the
model for studying a hypothetical chemical reaction
where the chemical concentrations oscillate, and
Volterra proposed the model to explain the increase
of predator fish in the Adriatic Sea, currently it
becomes a standard model, and the simplest model
of predator-prey interaction. Often, it is discussed in
standard textbooks of ordinary differential equations
and mathematical modeling, e.g., [3], [5], [5]. For
three population models, the effect on
the population dynamics because of these
parametric changes in the systems was studied
within invariant surfaces and, in terms of stability,
about equilibrium points, [6].
In general, the model is in the interest of
ecology. The Nicholson formula which is derived
from Lotka-Volterra used to determine the area of
discovery if the percentage parasitism and adult
parasite density are known, to construct as
competition curve with relates the percentage
parasitism to the fraction of the area covered by the
parasites and as a component in a population model
in which it represents parasite action, [7]. The
Lotka-Volterra model also expresses the tropic
interaction which considers the feeding rate as
directly proportional to the product of the
magnitudes of consumer supply and food supply,
[8]. On the other hand, Lotka-Volterra assumed that
the response of the populations would be
proportional to the product of their biomass
densities, [9].
However, it is also in the interest of theoretical
physics where Lotka-Volterra method was regarded
as the first principle in deriving
hydrodynamic equations of motion from
the equations of motion of the constituent particles,
[10]. Advancement of the model has been conducted
by considering several mathematical and physical
aspects. Taylor and Crizer's modified Lotka-
Volterra model would be better than the classic
model if in a biological situation the population had
a non-linear effect on each other,
[11]. Modified Lotka-Volterra competition model
with a non-linear relationship between species,
crowding effect has also been considered. Under the
Lotka-Volterra competition equation with a
nonlinear weak crowding effect, a stable coexistence
of many species is plausible, [12], [13].
The effect of diffusion on the model has been
discussed in papers, among others, [14], [15],
[16]. Diffusion can make the system persistent
regardless of the patch dynamics without diffusion.
More recently, Slavik, [17], considered
the Lotka-Volterra model on graphs, if both species
can move along the edges of the same connected
graph G. In a more general model, we might
consider two different connected graphs G1, G2,
one for each species.
This paper considers a Lotka-Volterra model
with a source term. The source term is applied to
one variable of the model or both variables.
Considering the Lotka-Volterra model as the
interaction model of predator-prey, the source term
may act as a harvesting scenario that applied merely
to the prey, to the predator, or both. The right source
term is important for optimal harvesting of the
predator and/or the prey.
Understanding of the Lotka-Volterra model is
very important for applications in integrated farming
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systems, especially to determine maximum
production and optimality of the system, for
example in an integrated farming system consisting
of vegetables as prey and fish as predators, [18]. The
sustainability of integrated farming systems
is interesting to discuss. It is mainly because of the
diversity of species and the potential for synergy
from integrating crops with livestock. However, the
ability of this system to maximize food production
has not been widely discussed in the literature and
needs to be explored further, [19].
2 Lotka-Volterra Model
In this section a Lotka-Volterra model without
carrying capacity will be discussed. Let , be
functions of time that represent the populations of
prey and predator, respectively. Lotka-Volterra
model, then, is given in the form [4]
(1)
Parameter represents the intrinsic growth rate
of prey, represents the rate at which predators
destroy prey, is parameter for the rate at which
predator population increases by consuming prey,
and is the death rate of predators.
System (1) has two critical points, i.e., the trivial
equilibrium , and nontrivial equilibrium
. On the other hand, the Jacobian matrix
of system (1) is
(2)
The Jacobian matrix evaluated at has
characteristic equation
,
(3)
which gives eigenvalues
, and
.
Since and are greater than 0, this implies that
is not stable. In fact, it is a saddle point.
For equilibrium point , the Jacobian matrix is
in the form
(4)
The Jacobian has characteristic equation
(5)
That gives the eigen value
and
. Since and are greater than 0, so the
eigen values are imaginary numbers, and is a
stable point.
It now will evaluate the trajectory of points
governed by system (1). Applying basic calculus by
eliminating variable t from (1) one gets
(6)
Solving (6), one has
(7)
where is any constant.
Fig. 1 shows the graphics of (7) in
Cartesian coordinate for various values of . The
parameters , , , have been
applied. There are three curves of system (7)
representing . Each curve represents a
trajectory of point for . Each
trajectory is called an orbit. The orbits are closed
curves. It is observed that the larger the value of ,
the larger curve.
Fig. 1: Graphics of (7), .
For further issue, the evolution of x and y
corresponds to the value of V = 5, 6, 8 as shown in
Fig. 1 will be presented. Since the analytical
solutions of x and y are not straightforward, it will
be shown the numerical solutions. To do so, an
initial condition i.e., pairs of should
be chosen. These pairs must satisfy (7) for
corresponding value of V. Table 1 shows the pairs
with corresponding value V.
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Table 1.Value of and for corresponding V
V
x (0)
y (0)
5
0.6
0.3994110349
6
1
0.17630435981
8
1.5
0.06696007606
Applying initial values as presented in Table 1,
the solutions of x with respect to variable t for
various value of V are presented in Fig. 2.
Fig. 2: Solution curves for x of system (1), where
(red), (green), (blue).
Similarly, the solutions of y with respect to
variable t for various values of V are presented in
Fig. 3.
Based on Fig. 2 and Fig. 3, some observations
can be made. The larger value of V implies the
larger the amplitude of x and y. Similarly, the larger
the value of V, the larger the period of x and y.
Moreover, the period of x and y are the same, for the
same value of V.
For V = 5 it is observed that the amplitude of x is
approximately 3.5, while the one of y is around 2.7.
For V = 6, the amplitude of x is 2.2, on the other
hand the one of y is around 1.7. Finally, for E = 8,
the amplitude of x is 1.7 and the amplitude of y is
about 1.1.
For V = 5 it is observed that the periods of x and
y are around 3.72. This means that point
requires 3.72 units of time to complete
one cycle of the red curve of Fig. 1. For V = 6, the
period of x and y is approximately 4.08 units of
time. Finally, for E = 8, the period of x and y is
about 4.8 units of time.
Fig. 3: Solution curves for y of system (1), where
(red), (green), (blue).
Period of x as prey and y as predator in this
model may provide information when the
population prey and predator are at its maximum
and minimum. Hence, this may be applied to predict
the optimal harvesting.
3 Lotka-Volterra Model with
Carrying Capacity
Carrying capacity is the ability of an ecosystem to
support the life of organisms in it in a sustainable
manner. Or in other words, the carrying capacity is
the upper limit of population growth, because the
population can no longer be supported by existing
facilities, resources, and environment. The Lotka-
Volterra model in the previous section assumes that
the environment provides unlimited resources for
prey growth. In fact, they must compete among
themselves for the resources. This underlies the
addition of the environmental carrying capacity
factor to the Lotka-Volterra model in this section.
Lotka-Volterra model with carrying capacity is
in the form
(8)
with is a parameter related to the environment
carrying capacity of the prey population.
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System (8) has three equilibrium points, i.e., the
trivial equilibrium , and two nontrivial
equilibriums
and
.
The Jacobian of system (8) is
(9)
The Jacobian matrix (9) evaluated at has
characteristic equation
(10)
which gives eigenvalues
and
.
Since and are greater than 0, this implies that
is not stable. It is a saddle point.
Evaluating the Jacobian matrix (9) at equilibrium
point ,it has characteristic equation
(11)
Equation (11) gives eigenvalues
and
. Since , it implies
. But there are
three cases for
:
Case 1:
.
Case 2:
.
Case 3:
.
Case 1,
implies
. Hence, the
equilibrium is not stable. On the other hand, Case
2,
implies
and
. Therefore,
equilibrium is nothing else, but the equilibrium
, meaning that and are the same points.
Moreover,
. Case 3 will be discussed later.
Evaluating Jacobian matrix (9) at , it has
characteristics equation
(12)
where the roots are the eigen values
Since , base on the value of
and
, there are 3 cases to consider:
Case 1:
.
Case 2:
.
Case 3:
.
Case 1,
implies that
and
. Hence, it is a stable point. Case 2
results in the collapse of into . Moreover,
and
. Observe that the eigen
values are similar to the ones of Case 2 of .
Investigate is stable point. Finally, Case 3
implies
meaning that lying in the
fourth quadrant. Hence, the Case 3 is out of
discussion of this paper, since x and y must be non-
negative.
For case 1
, the orbit of model (8) cannot
be obtained analytically as straight forward as for
the previous model. Hence, it will be presented
numerically. Fig. 4 shows the phase portrait of
model (8) for the case
. Initial values are
presented in Table 2, and the parameters are
, , , , . Orbit of initial
value (1.5,0.066) is red, of (1.7,0.1) is green, and of
(2,0.4) is blue. Observe that all orbits tend to .
Table 2. Initial value
x(0)
y(0)
Initial 1
1.5
0.066
Initial 2
1.7
0.1
Initial 3
2
0.4
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Fig. 4: Phase portrait system (8)
Fig. 5 shows the solution of x corresponds to the
orbits of the same color as in Fig. 4. All solutions
tend to the same value, i.e., x about 0.8.
Fig. 5: Solution curves of x of system (8) correspond
to the orbits in Fig. 4.
Fig. 6 shows the solution of y corresponds to the
orbits of the same color as in Fig. 4. All solutions
tend to the same value, i.e., y about 0.6.
Fig. 6: Solution curves of y of system (8) correspond
to the orbits in Fig. 4.
Phase portrait for case 2,
, is presented in
Figure 7. The initial values of the orbits are
presented in Table 2. Orbit of initial value (1.5,
0.066) is red, of (1.7,0.1) is green, and of (2,0.4) is
blue. Observe that all orbits tend to which is the
same point as .
Fig. 8 shows solutions of x (green) and y (red)
where the initial condition is (2,0.4). In this case, x
tends to be a carrying capacity parameter, while y
becomes extinct.
Fig. 7: Phase portrait for
The solution of x (red) and y (green) for case 2
correspond to the initial value in Table 2 are
presented in Fig. 8.
Fig. 8: Solution curve x (red) and y (green) for case
2,
.
In its application to integrated farming systems,
this can be used by farmers in determining the right
time to renew the carrying capacity of the
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environment, so that the prey does not become
extinct. Extinction of prey can result in a decrease in
the number of predators or even extinction due to a
lack of food sources.
4 Lotka-Volterra Model with periodic
Harvesting of Predator
In a predator-prey system, in addition to the
interaction between prey and predators, it is possible
to harvest only the predators, prey only or both prey
and predators at the same time.
In this section, the Lotka-Volterra model with a
periodic harvesting of predators is considered. The
model is in the form
(13)
where is a harvesting function for
the predator, which is always positive, is a
parameter related to the number of harvesting and
is the harvesting period. Fig. 9 shows a phase
portrait of the system (13) for various initial values
based on Table 2 and , . The red,
green and blue curves use the initial values x(0) and
y(0) in Table 2 for initial 1, initial 2 and initial 3,
respectively. The solutions of x and y correspond to
the orbits in Fig. 9 are presented in Fig. 10 and Fig.
11, respectively.
The number of peaks of green and red curves in
Fig. 9 are 5, but the number of peaks of the blue
curve is 6.This means that the ‘period’ of the blue
curve is smaller than the ones of green and red
curves. Observe that the peaks (the troughs) are not
always at the same height. Similar phenomenon also
happens for the y solution curves displayed in Fig.
11.
Fig. 9: Phase portrait system (13), where the initial
values presented in Table 2
Fig. 10: x solution of system (13) for the initial
values presented in Table 2
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Fig. 11: y solution of system (13) for initial values
presented in Table 2
Periodic harvesting of predators does not make
prey or predators extinct. In each period the
maximum value decreases. The difference in
determining the initial value affects the maximum
population size for each period. This simulation is
important to support the decision making of adding
the initial value in the next period so that the
maximum population number achieved in the next
period remains the same as the initial period.
5 Lotka-Volterra Model with Periodic
Harvesting of Prey and Predator
The Lotka-Volterra model with a periodic
harvesting of prey and predator that depends on time
is represented by
(14)
where is a parameter related to the number of
harvestings of prey, is periodic time for x and
is a harvesting function for prey.
Phase portrait of the system (14), when
and the initial values of x and y are given
in Table 2. Phase portrait system (14) with
various initial values is shown in Fig. 12.The red,
green, and blue curves use the initial values x(0) and
y(0) in Table 2 for initial 1, initial 2 and initial 3,
respectively.
The x and y solution of system (14) correspond
to orbits presented in Fig. 12 are shown in Fig. 13
and Fig. 14, respectively. Similar to the previous
section, the number of peaks of green and red curves
in Fig. 13 and Fig. 14 are 5, but the number of peaks
of blue curves are 6.
Fig. 12: Phase portrait of system (14), for the initial
values given in Table 2
Fig. 13: x solution of system (14) with initial values
given in Table 2
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Fig. 14: y solution of system (14) with initial values
given in Table 2
Periodic harvesting of prey and predators
simultaneously can provide multiple benefits to an
integrated farming system, [20]. However, it can be
seen in Fig. 12 that the harvesting of prey and
predators at the same time can cause a decrease in
the number of prey and predator populations in the
following period.
A sustainable seed supply is one of the keys to
the success of an integrated farming system, [21], so
that the farmer can decide when to add seeds as an
initial value to get a maximum value of prey and
predators in the next period will be the same as the
previous period. The right time to add these seeds
can be predicted using the simulation of system
(14).
Diversification of production between
horticultural crops and livestock in an integrated
manner, sustainable supply of inputs, and efficient
use of natural resources are important factors in an
integrated farming system, [21]. The supply of
inputs to plants, in this case acting as prey, can be in
the form of the carrying capacity of the environment
which limits the development of prey. This is added
to the model which is discussed in the following
section.
6 Lotka-Volterra Model with Periodic
Harvesting of Predator and
Carrying Capacity
In systems (13) and (14) there is no self-competition
for x, while in the real world the growth of a
population is limited by the carrying capacity of the
environment. Therefore, this section discusses the
Lotka-Volterra model with harvesting on predators
with the presence of parameters related to the
carrying capacity of x population.
Modification of system (13) by adding the
carrying capacity of the environment that limits x is
obtained Lotka-Volterra model with periodic
harvesting of predator that depends on time may be
represented by
(15)
The initial value data from Table 2 are still
considered. Corresponding orbits of system (15)
with these initial values are displayed in Fig. 15. In
this figure and t = 0, 1...,100. The red, green
and blue curves use the initial values x(0) and y(0)
in Table 2 for initial 1, initial 2 and initial 3,
respectively.
The x solutions of the system (15) that
correspond to the orbits in Fig. 15 are presented in
Figure16. It can be seen in the portrait phase in Fig.
15 that the portrait phase leads to a fixed phase
trajectory. It can also be seen that in the solution
curve in Fig. 16 and Fig. 17, the solution curve at
about t between 20 and 40, the population size
decreases and starts to form the same curves.
To increase the population, carrying capacity
improvements are needed and can be predicted
using a simulation of the system (15) with parameter
values adjusted to the types of livestock and plants
that are integrated.
Fig. 15: Phase portrait system (15)
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Fig. 16: Solution curves for x of system (15), with
initial value in Table 2
Fig. 17: Solution curves for y of system (15), with
initial value in Table 2
7 Lotka-Volterra Model with Periodic
Harvesting of Prey and Predator
and Carrying Capacity
Lotka-Volterra model with periodic harvesting of
prey and predator that depends on time and carrying
capacity of the environment that limits x can be find
by modification of system (14) may be represented
by
(16)
Using the initial value data from Table 2,
and t = 0, 1,..., 100 and the other parameters
follow parameters of the previous section then
obtained phase portrait system (16) is shown in Fig.
18. It can be seen in the portrait phase in Fig. 18, the
portrait phase leads to a fixed phase trajectory. It
can also be seen that in the solution curve in Fig. 19
and Fig. 20, the solution curve at about t between 30
and 40 begins to form the same curve.
Periodic harvesting of prey and predators with
limited carrying capacity of prey may double
farmers' income, from prey harvesting and predator
harvesting. However, the decline in the number of
populations caused by the decrease of carrying
capacity of the prey population can be corrected so
that the prey and predator populations can increase
again. In this system, the difference in the initial
value affects the maximum population of the system
in each period, but at certain times it becomes the
same because of the influence of carrying capacity.
Fig. 18: Phase portrait system (16)
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Fig. 19: Solution curves for x of system (16), with
initial value in Table 2.
Fig. 20: Solution curves for y of system (16) with
the initial value in Table 2
Integration of several types of commodities may
increase agricultural productivity, profitability and
sustainability compared to the cultivation of one
commodity. The use of residues and by-products
adds value to integrated farming systems. Besides
that, an integrated farming system may increase
farm income and implement sustainable agriculture,
[22], [23].
System (16) can be used to determine when to
add carrying capacity and seeds as initial values so
that a sustainable integrated farming system can be
implemented.
8 Concluding Remarks and Open
Problem
This research has modified several variations of the
Lotka-Volterra models, namely modeling with
carrying capacity, periodic harvesting of predator,
periodic harvesting of prey-predator, periodic
harvesting of predator with carrying capacity, and
periodic harvesting of prey-predators with carrying
capacity. Numerical simulation is to describe the
consistency and behavior of these models using the
same initial value. Modeling with carrying capacity
illustrates that the higher the initial value for the
predator, the stability is achieved in a short time,
and vice versa. Modeling with periodic harvesting
of predators provides an overview of monitoring
harvest limits. This provides important information
because it maintains the sustainability of the
ecosystem. Harvesting criteria are needed, the
predators are harvested and remain consistent in the
ecosystem. Modeling for harvesting prey and
predator provides an indicator the optimum
harvesting is carried out, for example, harvesting in
trajectory points. Other information is to provide
scenarios for intervention of prey and predator. The
harvesting of predator with carrying capacity in a
steady state condition will still have interactions
between prey and predator as well as dynamically
based on the available capabilities. For periodic
harvesting of prey-predator and carrying capacity
modeling is sensitive to the initial value. These
models provide an overview of optimal harvesting
scenarios for further work. For example, we are
determining the precision of the initial value to
determine optimal yields.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
-Norma Muhtar (PhD student) carried out
mathematical modeling, analysis, and finishing the
paper.
-Edi Cahyono(promotor) is responsible for the main
idea of the research, and supervising the process.
-R. Marsuki Iswandi and Muhidin (co-promotors)
are responsible for the motivation of the research,
interpretation and future applications in agriculture.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2022.21.31
Norma Muhtar, Edi Cahyono,
R. Marsuki Iswandi, Muhidin
E-ISSN: 2224-2678
293
Volume 21, 2022
