Study of a Diseased Volterra Type Population Model featuring Prey

Refuge and Fear Influence

N. MOHANA SORUBHA SUNDARI1, S.P.GEETHA2,∗

1Department of Mathematics,

Chikkaiah Naicker College, Erode 638 004,

INDIA

2,∗Department of Mathematics,

Vellalar College for Women, Erode-638012,

INDIA

∗Corresponding author

Abstract: - In order to study the local stability characteristics of a predator-prey dynamical model, this work pro-

poses a Volterra-type model that takes into account the fear influence of prey resulting from predator domination.

Because of an outbreak of disease in the prey species, the prey gets classified as either healthy or diseased. Both

predator and prey species compete for their resources. In addition, the prey sought refuge against the predator.

All these factors are addressed when setting up the mathematical model. The biological validity of the model is

ensured by testing its boundedness. The equilibrium points have been identified. The short-term behavior of the

system is analyzed at all equilibrium points. Routh Hurwitz conditions are employed to examine the local stability

property.

Key-Words: - Predator-prey model, Equilibrium points, Routh Hurwitz conditions, Fear effect, Prey refuge,

Local stability.

Received: August 25, 2023. Revised: April 2, 2024. Accepted: April 26, 2024. Published: May 20, 2024.

1 Introduction

The framework existing among the living and non-

living organisms in the environmental structure ex-

hibit nonlinear feature. To investigate their interac-

tions, their behaviours are captured and mathemat-

ically described mainly in the form of differential

equations. Many predator-prey species demonstrate

an unexpected diversity of dynamical behavioural

patterns, which has sparked a boom in the design of

mathematical models of ecosystems.

The first predator-prey model has been devel-

oped by Alfred James Lotka and Vito Volterra. The

Lotka-Volterra system of equations has an extensive

record which originated before a century. These

equations expressed an association between two or

more species. From then on, various types of model

equations have been developed, modified, and ex-

tended extensively incorporating many traits of the

species under study. In [1], the authors focused on

the predator-prey system, striving to provide a cut-

ting edge over view of recent models incorporating

the Allee effect, fear effect, cannibalism and im-

migration, and juxtaposed the qualitative outcomes

achieved for each element with a special focus on

equilibria, both local and global stability and the pres-

ence of limit cycles. Anderson and May (1981) were

the first to explore a population model with infection.

Since then, many ecoepidemiological models have

been studied incorporating disease in prey / preda-

tor or both species with various modes of disease

transmission. Considering the Holling-type interac-

tion, [2], proved that a judicious selection of gen-

eral Holling parameters, disease management can be

achieved by regulating the interacting function within

the ecosystem. The authors in [3], studied a prey-

predator model where the disease spreads only among

predators, transmitted horizontally through contact

between infected and susceptible individuals. An epi-

demic model that integrates vertical and horizontal

transmission of infection employing a nonlinear in-

cidence rate was investigated, [4].

By offering refuge, the ecosystem provides some

kind of defense to the prey from predators. The in-

fluence of prey refuge was explored on the dynamic

behavior of the model using the Lotka-Volterra frame-

work that features a Holling type III functional re-

sponse, [5]. Prey refuges have highly complicated

consequences on the dynamics of population, [6]. The

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fear factor may alter the normal behavior of the prey,

which in turn has a significant impact on the popula-

tion model, [7], [8], [9], [10].

Fear induced by the risk of predation decreases

prey birth rates. Also limit cycle observation revealed

that predation fear can both stabilise and destabilise

the ecosystem, [11]. With an increasing fear effect,

the final density of the prey species may approach

zero, driving them to extinction. The fear phenom-

ena negatively impact prey survival, potentially a sig-

nificant factor in their extinction, [12], differing from

Wang’s result, [13]. The findings of [14], suggested

that increasing the prey refuge or Allee effect en-

hances the dynamic complexity of the system. More-

over, while the fear effect or the Allee effect does not

have an impact on the density of the predator, it can

reduce the predator population at a positive equilib-

rium, [14]. Fear can cause backward bifurcation and

chaos by supressing prey growth and disease trans-

mission leading to a significant reduction in the in-

fection rate, [15]. Ecologically, prey adapt to fear be-

yond a critical threshold, which is essential to sustain

the ecosystem. Fear not only stabilizes the system, but

also regulates disease and diminishes predator pop-

ulation, [16]. Increasing fear level enhances system

stability by eradicating periodic solutions and reduc-

ing predator population at the coexisting equilibrium

point without leading to predator extinction. Also,

prey refuge significantly contributes to predator per-

sistence, [17]. The fear factor serves to stabilize the

dynamics of the system, [18]. Despite its study, the

influence of predator fear on prey with the inclusion

of disease in the dynamical model has not yet been

fully addressed, [19], [20], [21].

The objective of the present work is to formu-

late and study a predator-prey model that integrates

predators’ fear effects on prey together with prey af-

fected by disease and with a Volterra-type functional

response. The sections in this paper are ordered as

follows. Section 2 presents the mathematical popula-

tion model. Section 3 discusses the positivity of the

solution and the boundedness. In section 4, all the

equilibrium points are determined. Section 5 analysis

the local stability behavior at the equilibrium points.

The last section is the conclusion.

2 Mathematical Model

The proposed mathematical dynamical model con-

sists of the prey density Xand the predator density

zat any time t. As a result of infection in the prey

group, they are classified as susceptible prey xand

infected prey y. Only susceptible prey reproduces.

There is intraspecific competition in the prey and also

in the predator species. The predator preys on suscep-

tible and infected prey. Without predator, the prey

species increases logistically. The prey is the only

source of food, and in conditions of nonavailability

of prey, the predator dies. With these assumptions the

model takes the form:

˙x=α

1 + fz x−δ1x2−δ2xy −d1x−β1xy

−c1(1 −m)xz =F1(x, y, z),(1)

˙y=β1xy −δ3y2−δ4xy −d2y

−c2(1 −m)yz =F2(x, y, z),(2)

˙z=µ1c1(1 −m)xz +µ2c2(1 −m)yz

−δ5z2−d3z=F3(x, y, z),(3)

with initial conditions

x(0) ≥0, y(0) ≥0, z(0) ≥0.(4)

All the parameters are presumed positive.

Nomenclature

xsusceptible prey density

yinfected prey density

zpredator density

ffear rate of the prey

αgrowth rate of prey

β1disease transmission rate

δ1competition within the susceptible prey

δ2competition between the susceptible

and infected prey

δ3competition within the infected prey

δ4competition between the susceptible

and infected prey

δ5competition within the predator

c1catchability rate of susceptible prey

c2catchability rate of the infected prey

µ1conversion rate of susceptible prey

µ2conversion rate of infected prey

m∈[0,1) constant proportion of prey taking refuge

3 Positivity and Boundedness

We prove that the system given (1)-(3) with (4) is well

posed mathematically in the positive quadrant

Ω = {(x, y, z) /x≥0, y ≥0, z ≥0}

and solutions exists for all positive time. The vari-

ables x, y, z represent biological species and have the

domain Ωin R3

+. The R.H.S of system (1)-(3) is con-

tinuously differentiable and locally Lipschitz in the

first quadrant Ω. Hence, there are solutions for the

initial value problem (1)-(3) with non-negative initial

conditions.

Theorem 3.1 For every solution of system (1)-(3)

that starts in the positive quadrant, the solutions are

uniformly bounded.

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Proof: Considering (x, y, z)as the solution of (1)-(3)

with (4).

Let S=x+y+z.

Taking the time derivative of S, we get

S= ˙x+ ˙y+ ˙z

=α

1 + fz x−δ1x2−(d1x+d2y+d3z)

−δ2xy −δ4xy −δ3y2−δ5z2

−c1(1 −m)xz (1 −µ1)

−c2(1 −m)yz (1 −µ2)

Let η=min {d1, d2, d3},

After simple algebraic simplification,

S≤αx −δ1x2−ηS −(δ2−δ4)xy −δ3y2

−δ5z2−(1 −m)xzc1(1 −µ1)

−(1 −m)yzc2(1 −µ2)

If µ1<1, µ2<1, then the above equation becomes,

S≤αx −δ1x2−ηS

S+ηS ≤ −δ1x2−αx

δ1

Rearranging and writing as perfect squares,

S+ηS ≤ −δ1x−α

2δ12

+α2

4δ1

Let α2

4δ1=N, then ˙

S+ηS ≤N

By a theorem of differential inequality,

lim

|{z}

t→∞

supS (t)≤N

η,∀t > 0

Hence, the proof is complete.

4 Determination of Equilibrium

Points

Here the equilibrium points of system (1)-(3) are de-

termined.

1. f0(0,0,0) is the trivial equilibrium point.

2. f1(x, 0,0) is the infection free and predator free

equilibrium point where

x=α−d1

δ1

, α −d1>0

3. f2x, y, 0-the predator free equilibrium point

exists only with the existence of positive solution to

the below equations:

αx −δ1x2−δ2xy −d1x−β1xy = 0,

β1xy −δ3y2−δ4xy −d2y= 0.

From the above equation, we get,

y=(β1−δ4)x−d2

δ3

Then,

x=(α−d1)δ3+d2(δ2+β1)

δ1δ3+ (δ2+β1) (β1−δ4).

This equilibrium point exists if

x > d2

β1−δ4

,(β1−δ4)>0and (α−d1)>0.

4. f3(ex, 0,ez)- the equilibrium point without

disease exists only with the existence of positive

solution to the following equations:

1 + fez−δ1ex−d1−c1(1 −m)ez= 0,(5)

µ1c1(1 −m)ex−δ5ez−d3= 0.(6)

From (6), we get,

ex=δ5ez+d3

µ1c1(1 −m)(7)

Substituting (7) in (5) and letting A=δ1

µ1c1(1−m), we

obtain

[c1(1 −m)f+Aδ5f]ez2

+ [c1(1 −m) + Aδ5+ (Ad3+d1)f]ez

+ (Ad3+d1)−α= 0 (8)

Utilizing sign rule of Descarte’s if (Ad3+d1)< α,

then (8) has a unique positive root.

5. f4e

ex, e

ey, e

ez- the interior equilibrium point ex-

ists only with the existence of positive solution to the

below equations:

1 + fe

ez−δ1e

ex−δ2e

ey−d1−β1e

ex−c1(1 −m)e

ez= 0,

(9)

β1e

ex−δ3e

ey−δ4e

ex−d2−c2(1 −m)e

ez= 0,

(10)

µ1c1(1 −m)e

ex+µ2c2(1 −m)e

ey−δ5e

ez−d3= 0.

(11)

From (11) we have

ez=µ1c1(1 −m)e

ex+µ2c2(1 −m)e

ey−d3

δ5

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Let µ1c1(1−m)

δ5=w1;µ2c2(1−m)

δ5=w2;d3

δ5=w3.

Then,

ez=w1e

ex+w2e

ey−w3(12)

exists if w1e

ex+w2e

ey > w3(13)

Substituting e

ezin (10), we get,

ey=[(β1−δ4)−c2(1−m)w1]

x+[c2(1−m)w3−d2]

c2(1−m)w2+δ3.

Let (β1−δ4)−c2(1 −m)w1=A1;

c2(1 −m)w3−d2=A2;

c2(1 −m)w2+δ3=A3.

Therefore, e

ey=A1

x+A2

A3exists, provided that

A1e

ex+A2>0.

(12) can be written as

ez=w1+w2A1

A3e

ex+w2A2

−w3.

Substitute e

eyand e

ezin (9), we get

−[δ1fw1+w2A1

A3+(δ2+β1)fA1

A3w1+w2A1

A3

+c1f(1 −m)w1+w2A1

A32]e

ex2

−[δ1+(δ2+β1)A1

A3+δ1fw2A2

A3−w3

+(δ2+β1)fA1

A3w2A2

A3−w3

+(δ2+β1)fA2

A3w2A1

A3+w1

+ (d1f+c1(1 −m)) w2A1

A3+w1

+2c1f(1 −m)w2A1

A3+w1w2A2

A3−w3]e

+α−[(δ2+β1)A2

A3+d1+(δ2+β1)fA2

A3w2A2

A3−w3

+ (d1f+c1(1 −m)) w2A2

A3−w3

+c1f(1 −m)w2A2

A3−w32] = 0

Using the sign rule of Descarte’s, e

exhas a unique

positive root if

[(δ2+β1)A2

A3+d1+(δ2+β1)fA2

A3w2A2

A3−w3

+ (d1f+c1(1 −m)) w2A2

A3−w3

+c1f(1 −m)w2A2

A3−w32]< α.

5 Analysis for Local Stability

Behavior

Determining the local stability of the equilibrium

points involves analyzing the behavior of the sys-

tem near each equilibrium point. This analysis typ-

ically involves linearizing the system of differential

equations around each equilibrium point and examin-

ing the eigenvalues of the resulting Jacobian matrix.

The sign of the real parts of the eigenvalues indicates

whether the equilibrium point is stable, unstable, or

semi-stable. The necessary criteria for the system (1)-

(3) to be stable locally at the equilibrium points are

determined in this section using the Routh Hurwitz

criterion, [22].

Theorem 5.1 The trivial equilibrium point

f0(0,0,0) is locally asymptotically stable for

system (1)-(3) if d1> α.

Proof:

The Jacobin matrix at f0(0,0,0) is:

J(0,0,0) ="α−d10 0

0−d20

0 0 −d3#

If d1> α, then all the roots of the characteristic equa-

tion of J(0,0,0) are negative and thus the trivial equi-

librium point is locally asymptotically stable.

Theorem 5.2 The equilibrium point f1(x, 0,0) is lo-

cally asymptotically stable for the system (1)-(3) if

β1−δ4<d2δ1

α−d1and (α−d1)µ1c1(1 −m)< d3δ1.

Proof:

The Jacobin matrix at f1(x, 0,0) is:

J(x,0,0) =







−(α−d1)−(δ2−β1)(α−d1)

δ1−(α−d1)[αf+c1(1−m)]

δ1

0(β1−δ4)(α−d1)

δ1−d20

0 0 (α−d1)µ1c1(1−m)

δ1−d3







If β1−δ4<d2δ1

α−d1and (α−d1)µ1c1(1 −m)<

d3δ1, then all the roots of the characteristic equation

of J(x,0,0) are negative. So, the equilibrium point

f1(x, 0,0) is locally asymptotically stable.

Theorem 5.3 The equilibrium point f2x, y, 0is lo-

cally asymptotically stable for the system (1)-(3) if it

satisfies the condition

(α−d1)<2δ1x+ (δ2+β1)y, (14)

(β1−δ4)x < 2δ3y+d2,(15)

µ1c1(1 −m)x+µ2c2(1 −m)y < d3.(16)

Proof:

The Jacobin matrix at f2x, y, 0is:

J(x,y,0) ="c11 c12 c13

c21 c22 c23

c31 c32 c33#

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where

c11 = (α−d1)−2δ1x−(δ2+β1)y;

c12 =−(δ2+β1)x;

c13 =−[αf +c1(1 −m)]x;

c21 = (β1−δ4)y;

c22 = (β1−δ4)x−2δ3y−d2;

c23 =−c2(1 −m)y;

c31 = 0; c32 = 0;

c33 =µ1c1(1 −m)x+µ2c2(1 −m)y−d3.

The characteristic equation of J(x,y,0) is

(c33 −λ)[λ2−(c11 +c22)λ+c11c22 −c12c21] = 0.

(17)

ie., (c33 −λ) = 0,

(18)

and λ2−(c11 +c22)λ+c11c22 −c12c21 = 0.

(19)

From (18) λ=c33 <0provided µ1c1(1 −m)x+

µ2c2(1 −m)y < d3.

Using Routh-Hurwitz criterion (19) has negative

roots only if conditions (14) and (15) are satisfied.

Henceforth, the theorem follows.

Theorem 5.4 The equilibrium point f3(ex, 0,ez)is lo-

cally asymptotically stable for the system (1)-(3) if it

satisfies the condition:

1 + fez<2δ1ex+d1+c1(1 −m)ez,

(20)

ex < min d2+c2(1 −m)ez

β1−δ4

,2δ5ez+d3

µ1c1(1 −m).

(21)

Proof:

The Jacobin matrix at f3(ex, 0,ez)is given by

J(e

x,0,e

z)="c11 c12 c13

c21 c22 c23

c31 c32 c33#

where

c11 =α

1+fe

z−2δ1ex−c1(1 −m)ez;

c12 =−(δ2+β1)ex < 0;

c13 =−hαf

(1+fe

z)2+c1(1 −m)iex < 0;

c21 = 0; c22 = (β1−δ4)ex−d2−c2(1 −m)ez;

c23 = 0; c31 =µ1c1(1 −m)ez > 0;

c32 =µ2c2(1 −m)ez > 0;

c33 =µ1c1(1 −m)ex−2δ5ez−d3.

The characteristic equation of J(e

x,0,e

z)is

(c22 −λ)[λ2−(c11 +c33)λ+c11c33 −c13c31] = 0.

(22)

ie., (c22 −λ) = 0,

(23)

and λ2−(c11 +c33)λ+c11c33 −c13c31 = 0.

(24)

From (22) λ=c22 <0if condition (21) is met.

By Routh-Hurwitz criterion (24) has negative roots

only if conditions (20) and (21) are satisfied.

Henceforth, the theorem follows.

Theorem 5.5 The equilibrium point f4e

ex, e

ey, e

ezis

locally asymptotically stable for the system (1)-(3) if

it satisfies the condition:

1 + fe

ez<2δ1e

ex+d1+ (δ2+β1)e

ey+c1(1 −m)e

ez, (25)

(β1−δ4)e

ex < 2δ3e

ey+d2+c2(1 −m)e

ez, (26)

µ1c1(1 −m)e

ex+µ2c2(1 −m)e

ey < 2δ5e

ez+d3,(27)

ϕ1< ϕ2,(28)

τ1+τ2>0.(29)

Proof:

The Jacobin matrix at f4e

ex, e

ey, e

ezis:

z)="c11 c12 c13

c21 c22 c23

c31 c32 c33#

where

c11 =α

1+f

z−2δ1e

ex−d1−(δ2+β1)e

ey−c1(1 −m)e

ez;

c12 =−(δ2+β1)e

ex < 0;

c13 =−αf

(1+f

z)2+c1(1 −m)e

ex < 0;

c21 = (β1−δ4)e

ey > 0;

c22 = (β1−δ4)e

ex−2δ3e

ey−d2−c2(1 −m)e

ez;

c23 =−c2(1 −m)e

ey < 0;

c31 =µ1c1(1−m)e

ez > 0; c32 =µ2c2(1−m)e

ez > 0;

c33 =µ1c1(1 −m)e

ex+µ2c2(1 −m)e

ey−2δ5e

ez−d3.

The characteristic equation of J(

z)is given

λ3+a1λ2+a2λ+a3= 0 (30)

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Here

a1=−[c11 +c22 +c33],

a2=c22c33 +c11c33 +c11c22 −[c23c32

+c31c13 +c21c12],

a3=−c11c22c33 +c11c32c23 +c12c21c33 −c12c31c23

−c13c21c32 +c13c31c22.

Let

ϕ1=c11c32c23 +c12c21c33 +c13c31c22

−(c13c21c32 +c11c22c33)>0,

ϕ2=c12c31c23 >0.

From conditions (25)-(27), a1>0, a2>0.

Condition (28) gives a3>0.

Now,

a1a2−a3=−c2

11[c22 +c33] + c13c31[c11 +c33] +

c12c21[c11 +c22]−c2

22[c11 +c33]−2c11c22c33 +

c23c32[c22 +c33]−c2

33[c11 +c22] + c12c31c23 +

c13c21c32.

Let

τ1=−c2

11[c22 +c33]+c13c31[c11 +c33]+c12c21[c11 +

c22]−c2

22[c11 +c33]−2c11c22c33 +c23c32[c22 +

c33]−c2

33[c11 +c22] + c12c31c23 >0;

τ2=c13c21c32 <0.

If τ1+τ2>0, then a1a2−a3>0.

Then by Routh-Hurwitz criterion, all the roots of (30)

are negative. Henceforth, the theorem follows.

6 CRQFOXVLRQ

A predator prey population model comprising healthy

prey, infected prey and predator is developed integrat-

ing fear effect and refuge factors of the prey. The dis-

ease is transmitted from infected to susceptible prey

with linear incidence rate. The predator feeds on both

the susceptible and infected prey following Volterra

type predation. The positivity and boundedness of

the solution demonstrate that the developed system

behaves well biologically. Five equilibrium points

are located. Conditions for the existence of the equi-

librium points are elaborately discussed. Analyzing

the stability of each equilibrium point allows under-

standing of how the system responds to small pertur-

bations around that point. This information is crucial

for predicting the long-term behavior of the ecologi-

cal system and understanding its resilience to external

factors. Furthermore, distinct requirements are deter-

mined for the system’s local stability at each of the

equilibrium points.

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