Chaotic Dynamics of a Tri-Topic Food Chain Model
with Beddington-DeAngelis Functional Response in
Presence of Fear Effect
Surajit Debnath
University of Calcutta
Prahlad Majumdar
University of Calcutta
Susmita Sarkar
University of Calcutta
Uttam Ghosh ( uttam_math@yahoo.co.in )
University of Calcutta https://orcid.org/0000-0001-7274-7793
Research Article
Keywords: Fear effect, Beddington-DeAngelis functional response, Global stability, Bifurcations, Di-
rection of Hopf bifurcation, Chaos
Posted Date: June 10th, 2021
DOI: https://doi.org/10.21203/rs.3.rs-596219/v1
License: This work is licensed under a Creative Commons Attribution 4.0 International License.
Read Full License
Version of Record: A version of this preprint was published at Nonlinear Dynamics on September 21st,
2021. See the published version at https://doi.org/10.1007/s11071-021-06896-0.
Nonlinear Dynamics manuscript No.
(will be inserted by the editor)
Chaotic dynamics of a tri-topic food cha in model with
Beddington-DeAngelis functional response in presence of fear eff ect
Surajit Debnath · Prahlad Majumdar · Susmita
Sarkar · Uttam Ghosh
Received: date / Accepted: date
Abstract The most important fact in the field of theoretical ecology and evolutionary biology is the strat-
egy of predation for predators and t h e avoidance of prey from predator attack. A lot of experimental works
suggest that the reduction of prey depends on both direct predation and fear of preda ti o n . We explore the
impact of fear effect and mutual interference into a three-species food chain model. In this manuscript, we
have considered a tri-topic food web model with Beddingt o n - DeA n ge li s functional response between inter-
acting sp ec ies , incorporating the reduction of prey and intermediate predator growth because of the fear
of intermediate and top predator respectively. We have provided parametric conditions on the existence of
biologically fe asib l e equilibria as well as their local and g lo b a l stability also. We have established conditions
of transcritical, saddle-node and Hopf bifurcation in vicinity of different equilibria. Finally, we performed
some numerical investig a t io n s to justify analy ti c a l find ing s.
Mathematics Subje c t Classific ati on : 39A30, 92D25, 92D5 0.
Keywords : Fear effect; Beddingt on - DeA n g el is functional response; Global stabili ty; Bifurcations; Di-
rection of Hopf bifurcation; Chaos.
1 Introduction
Analysis of dynamical activities of prey-predator interaction is one of the momentous themes fo r researchers
in mathematical biology and theoretical ecology. Uniform existence and universal importance make the
prey-predator interplay an attracti ve field of investigation. Prey-predator interp lay becomes the focus of
theoretical ecologist an d experimental biol og i st from the last few decades [
1,2]. Many mathemati ca l mod-
els of prey-predator interactions have been formulated and considered to investigate the consumption and
survival dynamics of species [3,4]. Prey-predator interactions may be governed by ordina ry differential
equations [3, 5], fractional differential equations [6,7], partial differential equations [4,8] and stochastic dif-
ferential equations [9], delay d iff erential equations [10] and difference equations also. These mathematical
models a re constructed and studied to identify di ffe rent environmental effects such as the interaction be-
tween species, Allee effect, refuge of species, harvesting, pattern formation, environmental fluctuations etc.
In the early n i n et eenth century Malthus first formulated prey-predator interaction through mathemati-
cal models [11,12]. The celebrated Lotka-Volterra mode l was eventually enhanced by introducing logistic
growth function for prey species [13,14], incorporating various functional response and environmental effects
and these d evelopments makes prey-predator interplay more and more realistic [15,16, 17]. The dynamics of
prey-predator interaction can significantly b e affected in presence of fear effect in the field of environmental
biology and ecology [
18].
The presence of predator populations has the most impact on a prey-predator system t h ro u g h direct
predation as well as fea r of predation. A large number of mathematical models have been concerned
about considering the direc t preda t io n on ly [19 , 20 , 2 1 , 2 2 , 2 3 ] and referenc e there in . Thos e models are con-
structed with only prey dependent, both monotone and non-monotone functional responses [15,16,20,24,
Uttam Ghosh
Department of Applied Mathematics, University of Calcutta , Kolkata-700054, India
E-mail:
uttam math@yahoo.co.in
25], both prey and predator dependent functional resp on s e like Cowley-Martin type functional response
[17], Beddington-DeAngelis type functional response [26,27], Monod-Haldane type functional response [28]
and ratio-dependent functional response [
20,29]. In 2009, Takeuchi et al. studied the d i ss en si o n along with
investing time on taking care of juvenile and searching for food/nutrients of mature prey in absence of
direct predation, while they assumed that matures accommodate their parental care time via learning [30].
Krivan (200 7 ) investigated that exchange among foraging and predation based on classical Lotka-Volterra
prey-predator system where either pre y or predator or both species were adaptive to maximize their inde-
pendent sturdin ess [31] .
The second one is th a t in presence of predator species, prey species may gent ly modify thei r behaviour
because of the fear of predation threat. Appearanc e of predator may influence prey species more effectively
than direct predation [
32,33,34,35,36]. The birth rat e of prey species has been subjected to modification
through the introduction of fear effect [18,37]. A lot of field observation suggests including the cost of fear
in a prey-predator system, not only direct preda ti o n [33,34,35,36]. It also may be the cause of anti-predator
behaviour for prey species which plays a signi fi c ant responsibility on demography of p rey species [38]. For
example, the reproduction physiology of elks (Cervus elaphus) influenced by wolves (Canis lupus ) in Greate
Yellowstone Ecosystem [
39].
Prey species may transform their habitat from higher-risk zone to l ower risk zone, to reduce the predation
rate [ 3 5 ]. In 2011, Zanette et al. [36] observed during a whole breedin g season that song sparrows (Melospiza
melodia) reduces their reproduction due to fear of predators in their offspring season. This reduction is
being occurred because of their anti-predator behaviour which persuades growth rate, in addition to their
offspring endurance ra t es since female song sparrows la i d a few eggs. Only some of those eggs can survive
while most of the nestlings perished in a nest. They also observed that a variety of anti-predator behaviour
is responsible for this effect. For example, frightened parents suckled their nestlings less, their nestlings
were lighter and much more likely to perish. Correlational affirmation for birds [
33,40,41,42], Elk [39],
Snowshoe hares [43] and dugongs [44] also d is p lay some indication that fear can interrupt prey-predat or
interactions. Very recently Elliott et al., [45] studied some field experiments on Drosophila mel a n og as t er as
prey and mantid as their predator species, to observe the effect of fear on populations ro b u s tn e ss in relation
to species density. They explored t h a t in p res en c e of ma ntid, the reproductive rate of drasophila reduces
in both their breeding in addition to non-breeding season s .
Depending on field experimental data, Wang et al., Zanette et al. [
18,36] developed mathematical formula-
tion for prey-predator interaction by introducing the cost of fear for prey because of predator species, where
fear shows a crucial function on prey birth rate. They also observed that strong anti-predator behaviour or
correspondingly most important cost of fear may reduce the risk of the existence o f oscillatory behaviours
and thus eliminate th e scenarios “paradox of enrichment”. They also displayed that cost of fear can stabilize
the system by eliminating periodic behaviour as observed in prey-predator interactions. Also, periodic os-
cillations may arise, emerging from either sub-critical or supercritical Hopf bifurcation under comparatively
lower co st for fear [
18]. Thus, the effect o f fear can produce multi-stability in prey-predator interplays. In
2017, Wang and Zou formulate a stage-structured prey-predator interaction with adaptive avoidance for
predator species by including fear of predator for prey species [37]. They divided the prey population into
juvenile and mature stage and constitute a system of delay differential system with maturat io n delay. Das
and Samanta [
46] analysed a stochastic prey-predator system with the effect of fear due to predator on
prey species and additional food is provided for a predator. Recently, Panday et al. [47] analy sed a tri-topic
food web system con s id erin g the effect of fear of top predator and intermediate predator on reproduction
of intermediate predator and prey species respectively. Here, they study the cost of fear on the stability
dynamics of the model system.
In th is paper, the Hastings–Powell model [
48] ha s been modified incorporating fear of intermediate and top
predator to prey and intermediate predator respectively with B e d d ing t o n –De An g el is functional response
for b o t h species. The organization of the manuscript is as follows: in the next segment, we have formulated
the model syst em. The basic dynamical results such as positivity, boundedness property and persistence
of model are provided in section 3. Biologically feasible equilibria of the model system and parametric
conditions of local and global stability are determined in section 4. Section 5 is dedicated to studying
transcritical bifurcation at an axial equilibrium point and conditio n s for the occurrence of saddle-node and
Hopf bifurcation around coexistence equilibrium point along wit h stability direction of Hopf bifurcation.
In Section 6, we perform some numerical simulations to justify our analytical findings, which also sh ows
the roles of fear effect on the dynamics of prey-predator interactions. Fina ll y, in se ct i o n 7, we summarize
some biological indications from our analytical observation and possible future scope for upcoming research
works.
2
2 Model formulation
In this research article, our main objective is to investigate the change of basic dynamical behaviour of a
tri-topic food chain model in presence of fear due to the appearance of higher sp eci es . For this purpo se, we
start by considering t h e Hestings-Powell model [
48]. In 1991 Hest in g s and Powell proposed a continuous
tri-topic food chain model by considering more naturalist logistic growth function for prey species and
Holling type-II function a l response in the form,
dX
dT
= R
0
X(1 −
X
K
0
) −
R
1
XY
A
1
X + C
1
(1a)
dY
dT
=
E
1
R
1
XY
A
1
X + C
1
− D
1
Y −
R
2
Y Z
A
2
Y + C
2
(1b)
dZ
dT
=
E
2
R
2
Y Z
A
2
Y + C
2
− D
2
Z (1c)
with initial condition X(0) ≥ 0, Y (0) ≥ 0, Z(0) ≥ 0. X(T ), Y (T ) and Z(T ) are population densities o f
prey, middle/ i ntermediate predator and top predator population at any inst a nt of time T respectively
and desc rip t i o n of model parameters are given in Table
1. In 2018, Pandey et al. [47], incorporated effect
of fear in above considered tri-topic food chain model. They assumed that intrinsic growth rate of prey
population red u c es due to appearance of intermediate predator and modified intrinsic growth ra te of prey
population becomes φ
1
(K
1
, Y ) =
R
1
1 + K
1
Y
, which is a monotonically decreasing function of both K
1
and
Y . Similarly modified growth rate o f intermediate predator population is φ
2
(K
2
, Z) =
R
2
1 + K
2
Z
, which is
also a monotonically decreasing function of both K
2
and Z. K
1
, K
2
are level of fear parameters of prey and
middle preda to r population respectively. On behalf of above consi d era t io n they investigate the dynamics
of following modified tri-to p ic food chain model
dX
dT
=
R
0
X
1 + K
1
Y
(1 −
X
K
0
) −
R
1
XY
A
1
X + C
1
(2a)
dY
dT
=
E
1
R
1
XY
(1 + K
2
Z)(A
1
X + C
1
)
− D
1
Y −
R
2
Y Z
A
2
Y + C
2
(2b)
dZ
dT
=
E
2
R
2
Y Z
A
2
Y + C
2
− D
2
Z (2c)
Fear functions φ
1
(K
1
, Y ) and φ
2
(K
2
, z) satisfies following ecologica l hypot h es is :
1. φ
1
(0, Y ) = 1 and φ
2
(0, Z) = 1 ; i.e., if there is no anti-predator behaviours of prey and middle predator
species, then there will be no reduction in the birth rate of prey an d mid d le predat o r species.
2. φ
1
(K
1
, 0) = 1 a n d φ
2
(K
2
, 0) = 1; i.e., if middle o r top predator s pecies becomes zero, t h en there is no
reduction in the birth rate of prey and middle predator species.
3. lim
K
1
→∞
φ
1
(K
1
, Y ) = 0 and lim
K
2
→∞
φ
2
(K
2
, Z) = 0; i.e., if anti-predator behaviour is very large, then
prey an middle predator reproduction ultimate ly becomes zero.
4. lim
Y →∞
φ
1
(K
1
, Y ) = 0 and lim
Z→∞
φ
2
(K
2
, Z) = 0; i.e., if middle predator and top predator species is
very large, then respectively prey and middle predator reproduction reduces and ultimately goes to zero,
due to large anti-predator behaviours.
5.
∂φ
1
∂K
1
< 0 and
∂φ
2
∂K
2
< 0; i.e., reproduction of prey and middle predator species d ec rea s es with in c rea se
of anti-predator behaviours.
6.
∂φ
1
∂Y
< 0 and
∂φ
2
∂Z
< 0; i.e., reproduction of prey and middle species decreases with an increase of middle
predator and top predator species density respectively.
They discuss boundedness, positivity of system solutions and persistence of the considered model system.
Parametric conditions of local and global stability of different feasible equilibria are inves ti ga te d by them
also. However, in the study by Pandey et al.[
47], it was assumed t h a t prey-predator interplay depends only
on prey density alone and it is of type-II functional response. A huge number of detailed critical inspection
on ecology and physiological evidence suggests that competition within predator species might be g ood for
predator population under certai n cond i ti o n s in a deterministic environment [49].
To take account of the above observation, we have modified t h e model of Pandey et al. [47] by considering
the Beddington-DeAngelis type function response, d epending on both interacting species. Thus the modified
3
Symbol Definition Dimension Non-dimensional repres entation
T Time time t = R
0
T
X Prey density biomass x =
X
K
0
Y Intermediate predator density biomass y =
Y
K
0
E
1
Z Top predator density biomass z =
Z
K
0
E
1
E
2
R
0
Prey intrinsic growth rate time
−1
...
R
1
Maximum predation rate of intermediate predator time
−1
...
R
2
Maximum predation rate of top predator time
−1
...
K
0
Prey carrying capacity biomass ...
K
1
Fear level of prey biomass k
1
= K
0
K
1
E
1
K
2
Fear level of prey biomass k
1
= K
0
K
2
E
1
E
2
A
1
Handling time of middle predator biomass
−1
a
1
=
R
0
A
1
R
1
E
1
A
2
Handling time of top predator biomass
−1
a
2
=
R
0
A
2
R
2
E
2
B
1
Mutual interference among middle predators biomass
−1
b
1
=
R
0
B
1
R
1
B
2
Mutual interference among top predators biomass
−1
b
2
=
R
0
B
2
R
2
C
1
Environmental protection for prey dimensionless c
1
=
R
0
C
1
K
0
E
1
R
1
C
2
Environmental protection for middle predator dimensionless c
2
=
R
0
C
2
K
0
E
1
E
2
R
2
D
1
Intermediate predator mortality rate time
−1
d
1
=
D
1
r
0
D
2
Top predator mortality rate time
−1
d
2
=
D
2
r
0
E
1
Conversion efficiency of intermediate predator dimensionless ...
E
2
Conversion efficiency of top predator dimensionless ...
Table 1: Descripti on of system variables and system parameters with their dimensions and non-dimensional representation.
shape of above-consi d ered model with Beddington-DeAngelis type function response is as follows:
dX
dT
=
R
0
X
1 + K
1
Y
(1 −
X
K
0
) −
R
1
XY
A
1
X + B
1
Y + C
1
(3a)
dY
dT
=
E
1
R
1
XY
(1 + K
2
Z)(A
1
X + B
1
Y + C
1
)
− D
1
Y −
R
2
Y Z
A
2
Y + B
2
Z + C
2
(3b)
dZ
dT
=
E
2
R
2
Y Z
A
2
Y + B
2
Z + C
2
− D
2
Z (3c)
where B
1
, B
2
represents mutual interference among middle predators and top predators respectively.
To reduce numbe r of system parameters, we introduce following dim ens io n le ss variables x =
X
K
0
, y =
Y
K
0
E
1
,
z =
Z
K
0
E
1
E
2
and t = R
0
T . Then the above system reduces to following form:
dx
dt
=
x
1 + k
1
y
(1 − x) −
xy
a
1
x + b
1
y + c
1
:= F
1
(x, y, z), (4a)
dy
dt
=
xy
(1 + k
2
z)(a
1
x + b
1
y + c
1
)
− d
1
y −
yz
a
2
y + b
2
z + c
2
:= F
2
(x, y, z), (4b)
dz
dt
=
yz
a
2
y + b
2
z + c
2
− d
2
z := F
3
(x, y, z) (4c)
with the initial conditio n x(0) ≥ 0, y( 0 ) ≥ 0 and z(0) ≥ 0 and we introduce dimensionless parameters
as k
1
= K
0
K
1
E
1
, a
1
=
A
1
R
0
E
1
R
1
, b
1
=
B
1
R
0
R
1
, c
1
=
C
1
R
0
R
1
K
0
E
1
, d
1
=
D
1
R
0
, k
2
= K
0
K
2
E
1
E
2
, a
2
=
A
2
R
0
E
2
R
2
,
b
2
=
B
2
R
0
R
2
, c
2
=
C
2
R
0
R
2
K
0
E
1
E
2
and d
2
=
D
2
R
0
.
In next seg m ent of the manuscript, we shall establish boundedness, positivity of system solutions and
persistence of the system which will refer to that system is feasible, well-posed and exist for a long time.
4