The Long-Time Behaviour of a Stochastic Plankton Food Chain
Model
Liya Liu
1a
and Daqing Jiang
1,2 b
1
School of Petroleum Engineering, Key Laboratory of Unconventional Oil & Gas Development, China University of
Petroleum (East China), Ministry of Education, Qingdao, China
2
College of Science, China University of Petroleum, Qingdao, China
Keywords: Plankton Food Chain Model, Environmental Noise, Extinction, Ergodicity.
Abstract: This paper deals with problems of a tri-trophic plankton food chain model under environmental noise. When
considering plankton models, people are interested in when the plankton will be persist and extinct in a long
time. Sufficient conditions for the existence of an ergodic stationary distribution and extinction are
established. This work can predict the variation trend of the population of each species, so as to better
manage the population and provide theoretical basis for the safety and protection of the environment.
1 INTRODUCTION
1
In recent years, with the increasing population in
coastal areas and the rapid development of industry
and mariculture, coastal ecological environment has
been seriously damaged. The frequent occurrence of
red tide, sudden outbreak of fish diseases and
devastating death of large areas of cultured fish are
all related to the destruction of ecological
environment (Wang 2018). People have gradually
realized the importance of protecting marine
environment after paying a huge economic cost, and
they are eager to know and use the ocean from a
scientific perspective. Therefore, it has become a
scientific research strategy for coastal countries to
establish a mathematical model that can be applied
to marine ecological environment and predict the
balance and evolution of marine ecological system.
Phytoplankton play an important role in aquatic
ecosystem (Guillard 1975). Therefore it is necessary
to study the dynamic mechanism of the population
growth of phytoplankton.
At present, studying the phytoplankton
population growth by ecological model has become
a hot research topic. Scholars all over the world have
established and analysed a large number of different
types of ecological model (such as ordinary
a
https://orcid.org/0000-0003-2614-7628
b
https://orcid.org/0000-0003-4131-5757
differential equations, delayed differential equations,
differential equation of diffusion.) to describe the
phytoplankton population growth and diffusion
process (Abdallah 2003, Fang 2017, Ghosh 2016,
He 2017, Samanta 2013, Smith 2015). As far as we
know, the mathematical model on the tri-trophic
food chain has not been theoretically explored.
However, we believe that this study may open many
windows for population dynamics and require in-
depth research in this area.
In this work, we use the model of Hastings and
Powell (Hastings 1991) as a starting point to
construct a model that includes intraspecies
competition and white noise. The deterministic tri-
trophic food-chain model is as follows:
1
1
2
12
11
12
2
2
22
2
(1 ) ,
1
,
11
1
.
axydx
xx
dt b x
axy a yz
dy
dy cy
dt b x b y
ayz
dz
dz cz
dt b y
=−−
+
=−−−
++
=−−
+
(1)
The model describes the rates of change in
densities of a basal species (x), its predator (y) and a
top predator (z). It includes logistic growth of the
basal species and Holling Type II functional
responses. In a typical example,
x
,
y
and
z
might
Liu, L. and Jiang, D.
The Long-time Behaviour of a Stochastic Plankton Food Chain Model.
DOI: 10.5220/0011188400003443
In Proceedings of the 4th Inter national Conference on Biomedical Engineering and Bioinformatics (ICBEB 2022), pages 105-110
ISBN: 978-989-758-595-1
Copyright
c
2022 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
105
represent the population densities of phytoplankton,
herbivorous zooplankton and carnivorous
zooplankton, respectively. The parameters are as
follows:
1
1
1
axy
bx+
and
2
2
1
ayz
by+
are predation rates of the
two predators, respectively;
1
dy
and
2
dz
are
mortality rates of the two predators;
2
1
cy
and
2
2
cz
denote the density regulation of the two predators.
There are four nonnegative equilibria for system
(1):
0
(0,0,0)E
,
1
(1,0,0)E
,
211
(, ,0)Exy
and
****
(, ,)Exyz
, where
11
(, ,0)xy
and
***
(, ,)
x
yz
satisfy
11
1
11
11
11 1
11
1,
1
,
1
ay
x
bx
ax
cy d
bx
+=
+
−=
+
(2)
*
*
1
*
1
**
*
12
11
**
12
*
*
2
22
*
2
1,
1
,
11
,
1
ay
x
bx
ax a z
cy d
bx b y
ay
cz d
by
+=
+
−−=
++
−=
+
(3)
respectively.
2
E
is nonnegative equilibria if there is
positive solution of Equations (2) and
*
E
is a
positive equilibrium if there is a positive solution of
Equation (3). In this paper, we assume
2
E
and
*
E
always exist as nonnegative equilibria.
Environmental stochastic perturbations can affect
population dynamics inevitably. Real population
systems are always exposed to uncertain
environmental factors (Wen 2015, Zhao 2016).
Motivated by above facts, in this paper, taking into
account the effect of randomly fluctuating
environment, we assume that the parameters
involved in the model (1) fluctuate around some
average value. Thus we study a stochastic plankton
food chain model:
1
11
1
12
11 2 2
12
2
22 3 3
2
(1 ) ( ),
1
(),
11
(),
1
ay
dx x x dt xdB t
bx
ax az
dy y d c y dt ydB t
bx b y
ay
dz z d c z dt zdB t
by
σ
σ
σ
=−− +
+
=−−−+
++
=−−+
+
(4)
where
1
()Bt
,
2
()Bt
and
3
()Bt
are standard one-
dimensional independent Brownian motion, and
0
i
σ
>
are the intensity of the white noise,
1, 2, 3i =
.
2 GLOBAL DYNAMICS
When considering plankton models, we are
interested in when the plankton will be persist and
extinct in a long time. Here, we will talk about the
persistence and extinction of system (4).
2.1 Ergodicity
Ergodicity is one of the most significant
characteristics, meaning the stochastic plankton
model has a stationary distribution which shows the
survival of the plankton in the future. Now we state
sufficient conditions for the existence and
uniqueness of an ergodic stationary distribution of
the positive solutions to system (4) in the following
theorem.
Theorem 2.1. Assume that the following
conditions hold
*
11
*
1
1,
1
aby
bx
>
+
(5)
*
22
1
*
2
1
abz
c
by
>
+
(6)
and
2* 2 * *
121
2* **
31 2
*
*2
11
*
1
*
*2
22
1
*
2
***2
122
(1 )
22
(1 )(1 )
2
1(),
1
min ( ) , ,
1
(1 )(1 ) ( )
xbxy
bx b y z
aby
x
bx
abz
cy
by
bx b y c z
σσ
σ
+
++
++
−
+
<−
+
++
(7)
where
****
(, ,)Exyz
is the positive equilibrium of
system (1), then system (4) admits a stationary
distribution which is ergodic.
Proof. According to the analysis in (Liu 2019,
Zhu 2007), we only need to show there exists a non-
ICBEB 2022 - The International Conference on Biomedical Engineering and Bioinformatics
106
negative
2
C
-function
V
and a neighbourhood
U
such that
L
V
is negative for any
3
((), () () \,)
x
tytzt RU
+
∈
. Construct a
2
C
-
function
3
:VR R
+
→
in the following form
**
*
**
1
*
**
2
*
(, ,) log
+ log
+ log ,
x
Vxyz x x x
x
y
Ayy y
y
z
Azz z
z
=−−
−−
−−
(8)
where
*
11
1
A
bx=+
and
**
21 2
(1 )(1 )
A
bx b y=+ +
.
Making use of
ˆ
I
to s
′
formula and combing with
(3) lead to
**2
*2
11
*
11
*2
*2
1
11
**2 *2
122 1 2
*
22
*2
*2
23
22
*
*2
11
*
1
*
*2
22
11 22
*
2
*2
*2
1
()
()
(1 )(1 )
()
2
()
(1 )(1 ) 2
()
2
1()
1
()
1
()
2
aby x x
LV x x
bx bx
x
Ac y y
Aa b z y y Ay
by by
Az
Ac z z
aby
xx
bx
abz
Ac y y Ac
by
x
zz
σ
σ
σ
σ
−
=− − +
++
+− −
−
++
++
−−+
≤− − − −
+
−−−
+
⋅− +
*2
*2
23
12
*2 *2
12 3
*2
*2 *2
*2
23
112
22
:()( )
() ,
22 2
Az
Ay
mx x m y y m
Az
xAy
zz
σ
σ
σ
σσ
++
=− − − − −
⋅− + + +
(9)
where
123
,, 0mmm>
according to the conditions
(5) and (6).
Define a bounded closed set
{
3
(, ,) :
111
,, ,
UxyzR
xyz
εεε
εεε
+
=∈
≤≤ ≤≤ ≤≤
(10)
where
01
ε
<<
is a sufficiently small number. In
the set
3
\RU
+
, we can choose
ε
sufficiently small
such that the following conditions hold
*2
1
*
1
*2
*2 *2
23
112
1
()
4
,
22 2
mx
xm
Az
xAy
ε
σ
σσ
<−
++
(11)
*2
2
*
2
*2
*2 *2
23
112
1
()
4
,
22 2
my
ym
Az
xAy
ε
σ
σσ
<−
++
(12)
*2
3
*
3
*2
*2 *2
23
112
1
()
4
.
22 2
mz
zm
Az
xAy
ε
σ
σσ
<−
++
(13)
For convenience, we can divide
3
\RU
+
into six
domains,
{
}
{}
{}
3
1
3
2
3
3
(, ,) : ,
(, ,) : ,
(, ,) : ,
UxyzRx
UxyzRy
UxyzRz
ε
ε
ε
+
+
+
=∈<
=∈<
=∈<
3
4
3
5
3
6
1
(, ,) : ,
1
(, ,) : ,
1
(, ,) : .
UxyzRx
UxyzRy
UxyzRz
ε
ε
ε
+
+
+
=∈>
=∈>
=∈>
Obviously,
3
123456
\RUU U U U U U
+
=∪∪∪∪∪
. Next,
we will prove that
(, ,) 1LV x y z <−
for any
The Long-time Behaviour of a Stochastic Plankton Food Chain Model
107
3
(, ,) ( )\
x
yz R U
+
∈
, which is equivalent to
proving it on the above six domains.
If
1
(, ,)
x
yz U∈
, by (9), we have
*2
1
*2
*2 *2
23
112
**2
11
*2
*2 *2
23
112
*2
*2 *2
*2
23
112
1
(, ,) ( )
22 2
2()
22 2
1
()
2222
0,
LV x y z m x x
Az
xAy
mx m x
Az
xAy
Az
xAy
mx
σ
σσ
ε
σ
σσ
σ
σσ
≤− −
++ +
≤−
++ + ≤
−−++
≤
(14)
which follows from (7) and (11). Similarly, we
obtain
(, ,) 0LV x y z <
on
2
U
using the inequalities
(7), (12) and
(, ,) 0LV x y z <
on
3
U
according to
(7), (13).
If
4
(, ,)
x
yz U∈
,
5
(, ,)
x
yz U∈
or
6
(, ,)
x
yz U∈
, in view of (9), we get
(, ,) 0LV x y z <
.
Therefore, we have a conclusion that
3
(, ,) 0, (, ,) R .\
L
Vxyz xyz U
+
<∀ ∈
So we obtain that system (4) has an ergodic
stationary distribution. This completes the proof.
2.2 Extinction
We establish sufficient criteria for extinction of the
plankton in three cases. Before giving the main
result, we first give a lemma.
Lemma 2.1. Let
()
X
t
be the solution of the
stochastic differential equation
11
(1 ) ( ),dX X X dt XdB t
σ
=− +
(15)
then
()
X
t
converges weakly to distribution
ν
and
ν
is a probability measure in
R
+
such that
2
1
0
()1
2
udu
σ
ν
∞
=−
and its density is
()
1
22
1
()Qupu
σ
−
, where
2
1
1
2
1
2
2
1
1
2
1
2
1
2
Q
σ
σ
σ
σ
−
−
−
=Γ−
is a normal
constant and
22
11
22
()
u
p
uue
σσ
−
=
,
0u >
.
Since the proof is similar to (Liu 2012), Theorem
4.1 and we omit it here.
According to the Lemma, we get the following
result. For simplicity, we introduce the notations
0
1
() ( )
t
t
f
tfsds
t
=
and
w
→
means the
convergence in distribution.
Theorem 2.2. Let
( ( ), ( ), ( ))
x
tytzt
be the
solution of system (1.4) with any initial value
3
( (0), (0), (0))
x
yz R
+
∈
.
(i) If
2
1
1
2
σ
< , then all the plankton tend to zero
exponentially with probability one, i.e.,
lim ( ) lim ( ) lim ( ) 0
tt t
xt yt zt
→∞ →∞ →∞
===
a.s. (16)
(ii) If
2
1
1
2
σ
> and
2
2
11
0
1
()
2
udu
ad
au
σ
ν
∞
<+
+
, then
2
1
lim ( ) 1
2
t
t
xt
σ
→∞
=− a.s., ()
w
xt
ν
→ as t →∞,
2
2
11
0
1
log ( )
sup
()
lim
0,
12
t
yt
t
udu
ad
bu
σν
→∞
∞
≤
−+ <
+
a.s.
log ( )
sup 0lim
t
zt
t
→∞
<
a.s.
(17)
where
ν
is a probability measure in
R
+
which is
defined in Lemma 2.1. That is to say, the
phytoplankton
()
x
t
is persistent, while two kinds of
zooplankton
(), ()yt zt
go to extinction with
probability one.
(iii) If
2
1
1
2
σ
> ,
2
2
11
0
1
()
2
udu
ad
au
σ
ν
∞
>+
+
,
11 1
11
1
1
aby
bx
>
+
,
21
2
21
1
ay
d
by
>
+
and
ICBEB 2022 - The International Conference on Biomedical Engineering and Bioinformatics
108
1
22
2
21 2 11 11 1 2
21 21 1 11
2
3
2
(1 )
11 2(1)
,
2
ay a x bx y
by by c bx
d
σσ
σ
++
+
++ +
<+
then
2
1
lim ( ) 1
2
t
t
xt
σ
→∞
=− a.s., ()
w
xt
ν
→ as t →∞,
2
2
11
2
0
11 1
inf ( )
1()
0,
2
lim
t
t
yt
udu
ad
ac au
σ
ν
→∞
∞
≥
−+ >
++
2
3
21
2
21
1
22
2
211 1112
21 1 11
l
log ( )
sup
12
(1 )
0,
12(1)
im
t
ay
zt
d
tby
ax bxy
by c bx
σ
σσ
→∞
≤−+
+
++
+<
++
(18)
where
111
(, ,)
x
yz is the boundary equilibrium of
system (2) and
ν
is defined in Lemma 2.1. That is to
say, the phytoplankton
()
x
t
and herbivorous
zooplankton
()yt
are persistent, while carnivorous
zooplankton
()zt
go to extinction with probability
one.
The proof is quite similar to Theorem 4.1 in (Liu
2019), so we omit it here.
3 CONCLUSIONS
In this work, we use the model of Hastings and
Powell as a starting point to construct a more
realistic tri-trophic plankton food chain model (4)
that includes environmental perturbation. The
persistence and extinction of the plankton in a long
time are discussed.
Theorem 2.1 implies that the plankton will be
persistent, that is to say, all the phytoplankton and
zooplankton will survive in a long time under certain
conditions. Theorem 2.2 shows that the plankton
will extinct in three cases.
We have explored dynamic behaviors of the food
chain model. We believe that this work can predict
the variation trend of the population of each species,
so as to better manage the population and provide
theoretical basis for the safety and protection of the
environment.
ACKNOWLEDGEMENTS
This work was funded by the National Nature
Science Foundation of China (No 11871473), the
Fundamental Research Funds for the Central
Universities (No. 15CX08011A).
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