2.
SOME PROPERTIES OF THE
SOLUTIONS OF SYSTEM (2)
The
function in the right-hand side system (2) is continuous and has partial
derivatives on the space
. Therefore, system (2) satisfies
the Lipschitzian condition. Therefore, it has a unique solution. Further, the
time derivative of 
is zero when 
and
the time derivative of 
is
zero when
. Therefore, if the
solution of system (2) initiates at a non-negative point, then the components and
of the solution points of
system (2) cannot cross 
and 
of the solution points. Hence components
and P are always non negative.
From system (2), it gets
Therefore, the following Theorem
can be derived.
Theorem 1. Any
solution of system (2)initiate positively, satisfies the following:
1.
If 
, then 
.
2.
If 
, then 
and 
.
3.
If 
, then 
Note. The
first and the second part of the above theorem, tell us that all solution of system
(2) are
bounded,
while the third part makes clear that under condition , the prey species persist continuously.
3.
STEADY STATES AND THEIR
STABILITY ANALYSIS
System
(2) has the most three steady states. They are the total extinction steady
state
, which always exists, the
Predator-free steady state 
which exists, if and coexistence steady
state 
, where
and 
is a positive root of
where,
with 
From Theorem 1, we have 
. So, we search in 
Mean value theorem guaranteed that 
has a positive root if, 
and 
or and . Further, if 
, the coexistence steady
state exists.
To
investigate the locally asymptotical stability (LAS) and globally asymptotical stability
(GAS) for each steady state, firstly, let
linearize system (2) around a point 
Using the perturbed variables 
and
, system (2) can be
linearized as follows:
Where,
and 
i.
The total extinction steady
state
The
eigenvalues of 
, are 
and 
So, is LAS if and only if,
Further,
from theorem 1, it is proved that for any initial value of 
and
, if .
Therefore,
is GAS if and only if,
ii.
The predator-free steady state 
The
eigenvalues of 
, are
and
So,
is
LAS if and only if, .
Further,
the GAS for is given in the following theorem
Theorem 2. If
is exist, then it is GAS, if
(4)
(5)
Proof.
Consider the function
It is clear that 
is positive and 
,if and only if 
and . Further,
Accordingly,
Conditions (4) and (5) guarantee
that 
is
negative, this completes the proof.
iii.
The coexistence steady
state
The linearized system
around can be written as
(6)
Where,
, 
, 
,
, 
and
Theorem 3. If
is exist, then it is LAS if,
e
(7)
(8)
Proof.
Consider the function
It is clear that 
is positive and
, if and only if 
and
Due to conditions (7), (8),
it gets
This completes the
proof.
Theorem
4. If
exists, then it is GAS if,
(9)
(10)
Where, 
, 
and
Proof.
Consider the function 
where,
It is clear that 
and 
are positive and
, if and only if 
and
. Further,
+ 
and
So,
Conditions
(9) and (10) grantee that 
is
negative. This completes the proof.
4.
HOPF-BIFURCATION
The necessary condition for undergoing Hopf bifurcation near a
steady state point 
of system (2) is that,
the eigenvalues of 
+
are two complex
conjugate. Since and are the eigenvalues
of 
and and are the eigenvalues of 
so,
there
is no possibility to have a Hopf-bifurcation near and.
The
conditions that guarantee the occurring of Hopf-bifurcation near the
coexistence steady state 
are established in the
following theorem.
Theorem
5.
If
is exists and the following
conditions hold:
and 
(11)
(12)
(13)
then,
at 
,System (2) undergoes a
Hopf-bifurcation near
, where 
and 
are given in the proof.
Proof.
The eigenvalues of 
+
satisfy the equation
Clearly,
the roots of the above equation are neither zero nor positive. Therefore, the
eigenvalues are negative or complex. Note that when
, condition 9, guarantees
that all eigenvalues have negative real part. Suppose 
, 
is the root of the equation 30,
and is least positive number such that
,
Then
(14)
(15)
Putting
in the above two equations, then
adding and squaring them, the following equation get
It is obvious that under condition (12), Eq. 1, always has one
and only positive root, say.
From Eq. (15), it gets
(17)
Due to condition 13, Eq. 17 has much positive solution, let be least positive satisfy Eq. 15.Further, suppose
, then from Eq. 14 and
Eq. 15, it gets
, which is impossible
because 
, therefore 
,
The proof is completed.
5.
NUMERICAL COMPUTATION
In
this section, some numerical simulations were conducted by using the method of
modification Euler rule, with the help of MATLAB Program. The aim of numerical
simulation is to confirm the analytical finding observed in the previous
sections and discover the impact of fear, Allee effect, and time delay on the
dynamics of components of system (2). First, lets choose the parameter
values as follows:
Fig.1
shows that trajectory of system (2) approaches coexistence free steady state
point, and since the parameter values given by (18), they satisfy the global
stability condition in Theorem3. So Fig.1 confirms analytical result regarding
to stability condition of
.
Figure
1: the
phase portrait show that trajectory of system 2 approaches coexistence
steady state point,
when
and other parameter values are as given in (18).
To show the impact of time lag, fear and Allee effect, and
time lag on the dynamics of system 2. Lets solve system 2
with varying
, and fixed others as given in (18).
See Fig.2, Fig.2and Fig.3.
For the parameter
values in (18), the bifurcation value of time delay in Theorem, is 
, therefore, we solve
system (2) when the time delay varying from7 to 9 and fixed other parameter
values given in (18), the value of 
in range
, see Fig.2.
In
Fg.2, it has been shown that, dynamics of the system may
induce a transition from the stability situation to the state where the
populations oscillate periodically when the time delay value increases.
Figure
2: Illustration
of bifurcation diagram for system 2, when 
varies from 7 to 9
and
other parameters are fixed as in (18).
Figure
3: Illustration
of bifurcation diagram for system (2), when 
varies from 0 to10
and
other parameters are fixed as in (18).
In
Fig. 3, it has been discovered that, when 
increases,
the stability of coexistence steady limit value of both prey and predator
decreases, which means fear directly affects prey dynamics as well as
indirectly effects predator dynamics.
Figure
4
: Illustration of bifurcation diagrams for system 2, when 
varies from 0 to 10
and
other parameters are fixed as in (18).
In Fig.4, it has
been observed that, when 
increases, the limit value of prey density
increases too while the limit value of predator density decreases.
In
general, Fig.1 confirms the analytical results regarding to stability for the
coexistence steady state, Fig.2 discovers that the time delay may
induce a transition of the dynamics of system from the a stability situation to
the state where the populations oscillate periodically or vice versa. Fear
affects negatively on both prey and predator species, Fig.3 shows that the fear
affects negatively on both prey and predator species and Fig.4 demonstrates
that Allee effect for
predator reproduction has positive impact on the prey density
while it has negative impact on the predators.
CONCLUSION
In
this paper, a predator- prey model has been proposed. For derivation purposes
of the proposed model, it has been taken into account the time lag
corresponding to the gestation period and the effect that the fear of predators
has on prey and Allee
effect for predator reproduction. Firstly, it is proved that
the model solution is bounded and the prey species persist
continuously under the condition . It is explored
that the possible biological feasible steady states of system (2) are the total
extinction steady
state, the predator-free steady state, and coexistence
steady state. It is proved that the total extinction steady
state
is LAS and GAS if and only if, and the
Predator-free steady state is LAS if and only if the Local stability of both the
total extinction steady
and the Predator-free steady state are independent of fear
levels. Alle effect and time lags on LAS for but big value of fear may
destabilize Predator-free steady state for
dome initial values of species because the Predator-free steady
state
is GAS if,
and . According to coexistence
steady state, the analytical and numerical result show the
time delay may induce a transition of the dynamics of system from
the a stability situation to the state where the populations oscillate
periodically or vice versa, fear affect negatively on both prey and
predator species and Allee effect for
predator reproduction has positive impact on the prey
density, while it has negative impact on the predators.
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(1)
is rate of intrinsic growth of prey.
is the carrying capacity, 
is the consumption rate of prey by predator;
is the effect of capture rate; 
is
the conversion and m is rate of predator aggression.
(2)
, 
and
parameters are positive, their description are given in Table 1.
