The Influence of Density Dependent Death Rate of Predator Species to
the Lotka-Volterra Predator Prey System with Fear Effect
QIANQIAN LI, QUN ZHU, FENGDE CHEN
College of Mathematics and Statistics
Fuzhou University
No. 2, wulongjiang Avenue, Minhou County, Fuzhou
CHINA
Abstract: - A Lotka-Volterra predator prey system incorporating fear effect of the prey species and density de-
pendent death rate of predator species is proposed and studied in this paper. Local and global stability property of
the equilibria are investigated. Our study shows that the density dependent death rate of predator species has no
influence to the persistent or extinction property of the system. However, with the increasing of the density de-
pendent death rate, the final density of the predator species is decreasing and the final density of the prey species is
increasing. Hence, the increasing of the the density dependent death rate enhance the possibility of the extinction
of the predator specie. Numeric simulations show that too high density dependent death rate and too high fear
effect of prey species may lead to the extinction of the predator species.
Key-Words: Lotka-Volterra predator prey model; Stability; Fear effect; Density dependent death rate
Received: July 5, 2022. Revised: February 26, 2023. Accepted: March 13, 2023. Published: March 24, 2023.
1 Introduction
The aim of this paper is to investigate the dynamic be-
haviors of the following Lotka-Volterra predator prey
system incorporating fear effect of the prey and the
density dependent death rate of predator species
du
dt =r0u
1 + kv −du −au2−puv,
dv
dt =cpuv −mv −ev2,
(1.1)
where uand vare the density of prey species and
the predator species at time t, respectively. r0is the
birth rate of the prey species, dis the death rate of the
prey species, ais the density dependent death rate
of the prey species, m+ev is the death rate of the
predator species, obviously, it is density dependent; p
denotes the strength of interspecific between prey and
predator; cis the conversion efficiency of ingested
prey into new predators; kis the level of fear, which
is due to anti-predator behaviours of the prey.
Recently, Wang, Zanette and Zou[1] proposed
the following Lotka-Volterra predator prey system
incorporating fear effect of the prey
du
dt =r0uf(k, v)−du −au2−puv,
dv
dt =cpuv −mv.
(1.2)
The system admits three nonnegative equilibrium,
E0(0,0),E1(r0−d
a,0) and E2(u, v), where u=m
cp,
and vsatisfies
r0f(k, v)−d−au −pv = 0.(1.3)
Concerned with the global stability property of the
system (1.2), the authors obtained the following
result.
Theorem A. Assume that r0< d, then E0is globally
asymptotically stable; The boundary equilibrium E1
is globally asymptotically stable if r0∈(d, d +am
cp ),
and the unique positive equilibrium E2is globally
asymptotically stable if r0> d +am
cp .
It brings to our attention that in system (1.2),
the authors did not consider the influence of the
intra-competition of the predator species, though
such an assumption were adopt by many scholars
([1]-[22]) and it seems reasonable. We should also
pay attention to the other case. In the lack of food
situation, competitive of food resource will become
urgent, and those predators that less food maybe
driven to extinction, this leads to the increasing of
the death rate of predator species. Hence, many
scholars ([23]-[33]) also proposed the predator prey
system with density dependent death rate of predator
species. It bring to our attention that, to this day,
though there are many papers ([1]-[11]) investigated
the dynamic behaviors of the predator prey system
incorporating the fear effect of prey species, there
are still no scholars consider the influence of the
density dependent death rate to the system (1.2). This
motivated us to propose the system (1.1).
The aim of this paper is to investigate the dynamic
behaviors of the system (1.1), and to find out the
influence of the density dependent death rate of
predator species.
The rest of the paper is arranged as follows. We
will investigate the local and global stability property
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of the equilibria of the system (1.1) in Section 2 and
3, respectively, and then discuss the influence of
density dependent death rate of predator species in
Section 4. By applying the existence theorem for
implicit function, we discuss the influence of the fear
effect and the density dependent death rate of the
predator species. Numeric simulations are presented
in Section 5 to show the feasibility of the main
results. We end this paper with a brief discussion.
2 The existence and local stability of
the equilibria
Concerned with the existence of the equilibria of
system (1.1), we have the following result.
Theorem 2.1.System (1.1) always admits the triv-
ial boundary equilibrium E0(0,0) and if r0> d
holds, the predator free equilibrium E1r0−d
a,0
exists. Also, there exists a unique positive equilibrium
E2(u∗, v∗),if
r0> d +am
cp (2.1)
holds, where u∗=ev∗+m
cp and v∗is the unique
positive solution of the equation
A1v2+A2v+A3= 0,(2.2)
where
A1=ckp2+aek,
A2=cdkp +akm +cp2+ae,
A3=dcp −r0cp +am.
(2.3)
Remark 2.1. By introducing the density dependent
rate of the predator species, system (1.1) also admits
a prey free equilibrium E3(0,−m
e), since −m
e<0,
E3is lack of biological meaning, and we will not in-
vestigate it.
Proof of Theorem 2.1. The equilibria of system (1.1)
satisfy the equation
r0u
1 + kv −du −au2−puv = 0,
cpuv −mv −ev2= 0.
(2.4)
From the second equation of (2.4), one has v= 0 or
u=ev +m
cp .Substituting v= 0 to the first equation
of (2.4) leads to
r0u−du −au2= 0.(2.5)
Equation (2.5) has solutions u1= 0 and u2=
r0−d
a.Hence, system (1.1) admits the trivial equi-
librium E0(0,0), and if r0> d holds, the predator
free equilibrium E1r0−d
a,0exists.
Next, substituting u=ev +m
cp to the first equa-
tion of (2.4) and simplifying it leads to
A1v2+A2v+A3= 0.(2.6)
Under the assumption of (2.1), one could easily see
that A3<0, hence, (2.6) admits a unique positive so-
lution v∗, consequently, system (1.1) admits a unique
positive equilibrium E2(u∗, v∗).
The first equation of (2.4) has a solution u= 0,
substituting this to second equation of (2.4) leads to
−mv −d1v2= 0.(2.7)
Hence, system (1.1) admits the prey free equilibrium
E3(0,−m
d1
).Since −m
d1
<0,E3has no biologi-
cal meaning, and we will not investigate the stability
property of this equilibrium.
This ends the proof of Theorem 2.1.
Theorem 2.2. The trivial equilibrium E0(0,0) is lo-
cally asymptotically stable if
r0< d (2.8)
holds; If
d < r0< d +am
cp (2.9)
holds, the predator free equilibrium E1r0−d
a,0
is locally asymptotically stable; The positive equilib-
rium E3(u∗, v∗)is locally asymptotically stable if
r0> d +am
cp (2.10)
holds, i.e, the positive equilibrium is locally asymp-
totically stable as long as it exists.
Proof. The Jacobian matrix of the system (1.1) is cal-
culated as
J=J11 J12
J21 J22 ,(2.11)
where
J11 =r0
kv + 1 −d−2au −pv,
J12 =−r0uk
(kv + 1)2−pu,
J21 =cpu −2ev −m,
J22 =cpu −m−2d1v.
(2.12)
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Then the Jacobian matrix of the system (1.1) about
the trivial equilibrium E0(0,0) is
J(E0(0,0))
= r0−d0
0−m!.(2.13)
Under the assumption (2.8) holds, the eigenvalues of
J(E0)are λ1=r0−d < 0,λ2=−m < 0. Thus, the
trivial equilibrium E0(0,0) is locally asymptotically
stable.
It follows from (2.3) that the Jacobian matrix of
the system (1.1) about the predator free equilibrium
E1r0−d
a,0is
JE1(r0−d
a,0)
= −(r0−d)−(r0k+p)r0−d
a
0cpr0−d
a−m!.
(2.14)
Under the assumption (2.9) holds, the eigenvalues of
J(E1)are λ1=−(r0−d)<0,λ2=cpr0−d
a−m <
0. Thus, E1(r0−d
a,0) is locally asymptotically stable.
The Jacobian matrix of the system (1.1) about the
positive equilibrium E2(u∗, v∗)is
J(E2(u∗, v∗))
= −au∗−r0u∗k
(kv∗+ 1)2−pu∗
cpv∗−ev∗!.
(2.15)
Then we have
DetJ(E2(u∗, v∗))
=aeu∗v∗+cpv∗u∗r0k
(kv∗+ 1)2+p
>0
and
T rJ(E2(u∗, v∗)) = −au∗−ev∗<0.
So that both eigenvalues of J(E2(u∗, v∗)) have neg-
ative real parts, consequently, E2(u∗, v∗)is locally
asymptotically stable.
This ends the proof of Theorem 2.2.
3 Global asymptotical stability
Concerned with the global stability property of the
equilibria of system (1.1), we have the following re-
sult.
Theorem 3.1. Assume that r0< d, then trivial
equilibrium E0is globally asymptotically stable; The
predator free equilibrium E1is globally asymptoti-
cally stable if r0∈(d, d +am
cp ), and the unique pos-
itive equilibrium E2is globally asymptotically stable
if r0> d +am
cp .
Proof. We first show that the system admits no limit
cycle in the first quadrant. Let’s consider the Dulac
function B(u, v) = 1
uv ,then
∂(P B)
∂u +∂(QB)
∂v
=1
uv r0
kv + 1 −d−2au −pv
−1
u2vr0u
kv + 1 −du −au2−puv
+cpu −2ev −m
uv −cpuv −ev2−mv
uv2
=−au +ev
uv <0,
(3.1)
where P(u, v), Q(u, v)represent the two functions
on the right hand side of system (1.1). By Dulac The-
orem[34], there is no closed orbit in the first quadrant.
(1) When r0< d, the system admits only one non-
negative equilibrium E0(0,0), this, together with the
fact that the system has no periodic orbit in R+
2, im-
plies that every positive solution will approach E0,
that is, E0is globally asymptotically stable;
(2) When d<r0< d +am
cp , the system admits two
equilibrium E0and E1. Noting that in this case, E0
is unstable, and E1is locally asymptotically stable,
also, the system has no periodic orbit in R+
2hence,
every positive solution will approach E1, that is, E1
is globally asymptotically stable;
(3) When r0> d+am
cp , the system admits three equi-
libria E0,E1and E2, since in this case, only E2is lo-
cally asymptotically stable, while E0and E1are both
unstable. This, together with the fact that the system
has no periodic orbit in R+
2implying that every posi-
tive solution will approach E2, that is, E2is globally
asymptotically stable;
The proof of Theorem 3.1 is finished.
Remark 3.1. Compared with Theorem 3.1 and The-
orem A, one could see that the dynamic behaviors of
system (1.1) is similar to the dynamic behaviors of
system (1.2). The density dependent death rate of
predator species has no influence to the persistent and
extinction property of the system.
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4 The influence of parameters kand
e
Following we will discuss the influence of fear
effect and the density dependent death rate of predator
species.
Denote
F(u∗, v∗, k, e) = r0
1 + kv∗−d
−au∗−pv∗,
G(u∗, v∗, k, e) = cpu∗−m−ev∗.
(4.1)
Then the positive equilibrium E2(u∗, v∗)satisfies
(F(u∗, v∗, k, e)=0,
G(u∗, v∗, k, e)=0.(4.2)
By simple computation, we have
J=D(F, G)
D(u∗, v∗)=
Fu∗Fv∗
Gu∗Gv∗
=
−a−r0k
(1 + kv∗)2−p
cp −e
=ae +cpr0k
(1 + kv∗)2+p>0
for all u∗>0, v∗>0,k > 0, e > 0. Thus, the equa-
tions (4.2) satisfy the conditions of the existence the-
orem for implicit functions, then the equations (4.2)
determine two implicit functions of
u∗=u∗(k, e), v∗=v∗(k, e)
for all k > 0, e > 0.Also,
∂u∗
∂k =−1
J
D(F, G)
D(k, v∗),∂v∗
∂k =−1
J
D(F, G)
D(u∗, k),
∂u∗
∂e =−1
J
D(F, G)
D(e, v∗),∂v∗
∂e =−1
J
D(F, G)
D(u∗, e).
Since
D(F, G)
D(k, v∗)
=
−r0v∗
(1 + kv∗)2−r0k
(1 + kv∗)2−p
0−e
=er0v∗
(1 + kv∗)2>0,
D(F, G)
D(u∗, k)
=
−a−r0v∗
(1 + kv∗)2
cp 0
=cpr0v∗
(1 + kv∗)2>0,
D(F, G)
D(e, v∗)
=
0−r0k
(1 + kv∗)2−p
−v∗−e
=−v∗r0k
(1 + kv∗)2+p<0,
D(F, G)
D(u∗, e)
=−a0
cp −v∗
=av∗>0.
Hence, we have
(1)∂u∗
∂k <0, that is, the prey density u∗is a decreas-
ing function of k;
(2) ∂v∗
∂k <0, that is, the predator density v∗is a
decreasing function of k;
(3)∂u∗
∂e >0, that is, the prey density u∗is a increas-
ing function of e;
(4) ∂v∗
∂e <0, that is, the predator density v∗is a
decreasing function of e.
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5 Numeric simulation
We will introduce two examples to show the fea-
sibility of the main results.
Example 5.1. Let’s consider the following model
du
dt =4u
1 + kv −u−u2−2uv,
dv
dt =uv −v−ev2.
(5.1)
Here, corresponding to system (1.1), we take r0=
4, d =a=m= 1, p = 2,c= 0.5. then one could
see that
r0= 4 >2 = d+am
cp .(5.2)
Hence, it follows from Theorem 3.1 that for all
k∈[0,+∞)and e∈[0,+∞), system (5.1) admits
a unique positive equilibrium, which is globally
asymptotically stable. Numeric simulation (Fig. 1)
also supports this assertion.
Figure 1: Dynamic behaviors of the system (5.1),
the initial condition (u(0), v(0)) = (2,2),(2,1),
(2,0.2) and (2,0.5), respectively.
Now, let’s further take e= 1 in system (5.1), then
we could obtain the positive equilibrium
u∗(k) = 1 + v∗(k),
v∗(k) = −2k−3 + √4k2+ 36k+ 9
6k.
(5.3)
Fig. 2 shows that u∗(k)and v∗(k)both are the
decreasing function of k.
Figure 2: Relationship of u∗and kand v∗and k,
the red one is u∗(k), the blue one is v∗(k).
Also, by simple computation, we have
lim
k→+∞
v∗(k)
=lim
k→+∞
−2k−3 + √4k2+ 36k+ 9
6k
=lim
k→+∞
−2−3
k+ 2r1 + 9
k+9
4k2
6
=lim
k→+∞
−2−3
k+ 21 + 1
2·9
k+1
2·9
4k2
6
= 0,
lim
k→+∞
u∗(k)
=lim
e→+∞1 + v∗(k)= 1.
(5.4)
Now, let’s further take k= 1 in system (5.1), then
we could obtain the positive equilibrium
u∗(e) = 1 + e
2v∗(e),
v∗(e) = 1
2−e−4 + √e2+ 16e+ 32
e+ 2 .
(5.5)
Fig. 3 shows that u∗(e)is the increasing function of
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eand v∗(e)is the decreasing function of k.
Figure 3: Relationship of u∗and eand v∗and e,
the red one is u∗(e), the blue one is v∗(e).
Also, by simple computation, we have
lim
e→+∞
v∗(e)
=1
2lim
e→+∞−e−4 + √e2+ 16e+ 32
e+ 2
=1
2lim
e→+∞
−1−4
e+q1 + 16
e+32
e2
1 + 2
e
=1
2lim
e→+∞
−1−4
e+1+1
216
e+32
e2
1 + 2
e
= 0,
(5.6)
lim
e→+∞
u∗(e)
= 1 + 1
2lim
e→+∞
e−e−4 + √e2+ 16e+ 32
e+ 2
= 1 + 1
2lim
e→+∞
−1−4
e+q1 + 16
e+32
e2
1
e1 + 2
e
= 1 + 1
2lim
e→+∞
−1−4
e+1+1
216
e+32
e2
1
e1 + 2
e
= 3.
(5.7)
Numeric simulations in accordance with the the-
oretical analysis in Section 4.
6 Conclusion
Wang, Zanette and Zou[1] proposed a Lotka-
Volterra predator prey system incorporating the fear
effect of prey species, i.e., system (1.2). Their result
(Theorem A) indicates that the fear effect has no in-
fluence to the existence and stability of the equilibria.
Stimulated by the fact that the predator species may
also have nonlinear intra competition, and this may
lead to the density dependent death rate, we propose
the system (1.1), where both the fear effect and the
density dependent death rate are considered.
It seems interesting that the density dependent
death rate has no influence to the persistent or extinc-
tion of the system, since Theorem 3.1 shows that un-
der the same assumption of Theorem A, system (1.1)
admits the same dynamic behaviors as that of the sys-
tem (1.2). However, by applying the existence theo-
rem for implicit functions, we could show that u∗and
v∗both are the decreasing function of the k, that is,
with the increasing of the fear effect, the final den-
sity of predator and prey species decreasing. We also
show that u∗is the increasing function of eand v∗is
the decreasing function of the e, which means that the
density dependent death rate of the predator species
have negative effect on the final density of the preda-
tor species, and with the decreasing of the predator
species, the density of prey species become increas-
ing, since the chance of the prey species to be har-
vested becomes decreasing.
We mention her that despite the density dependent
death rate and fear effect have no influence to the per-
sistent property of the system, which seems similar to
that of the system (1.2). However, as was shown in
Example 5.2, v∗(k)is the decreasing function of k,
and v∗(k)→0as k→0. Hence, with the increasing
of the fear effect, the final density of predator species
approach to zero, which means the extinction of the
predator species. The reason for this maybe due to
the fact that increasing the fear effect may lead to the
decreasing of the density of prey species, and this fi-
nally leads to the lack of the food for maintain devel-
oping of the predator species. Example 5.2 also shows
that v∗(e)→0as e→+∞,and u∗(e)→u, where
u=r0−d
a, hence, with the level of the density
dependent death rate of predator species increasing,
the final density of predator species approach to zero,
which means the extinction of the predator species.
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Qianqian Li, Qun Zhu, Fengde Chen
E-ISSN: 2224-2678
336
Volume 22, 2023
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Contribution of individual authors to
the creation of a scientific article
(ghostwriting policy)
Qianqian Li wrote the draft.
Qun Zhu carried out the simulation.
Fengde Chen proposed the issue and revise the paper.
Sources of funding for research
presented in a scientific article or
scientific article itself
This work is supported by the Natural Science Foun-
dation of Fujian Province(2020J01499).
Conflict of Interest
The authors have no conflict of interest to
declare that is relevant to the content of this
article.
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: 10.37394/23202.2023.22.36
Qianqian Li, Qun Zhu, Fengde Chen
E-ISSN: 2224-2678
337
Volume 22, 2023
