INTRODUCTION
The study of interactions between species is a well researched field in ecology.
In the case of one species (the predator) feeding on another species (the prey),
a simple model was developed to understand the interaction. The model can be
written as a coupled system of two first order nonlinear differential equations
(Giordano et al., 2003). Non linear differential
equations are used in many fields (Taiwo and Abubakar, 2011;
Baghli and Benyettou, 2006; Prakash
and Karunanithi, 2009). The standard basic predatorprey model is the LotkaVolterra
system of equations (Rockwood, 2006). The major part of
existing theory on predatorprey interactions is built on this basic model.
Extensions of the basic model were developed to describe particular scenarios
and these involve the surroundings and the nature of the species involved. The
resulting system of equations describing the predatorprey equations has been
widely studied by many researchers (Naji and Balasim, 2007).
However, there is still a need to improve the present models. In nearly all
of these models, the main question that needs to be answered is whether the
species can coexist.
Relative to the size of the prey, the size of the predators might involve smaller
rate of change and this can result in competition amongst the predators (Mallah
et al., 2001; Akhtar and Khaliq, 2003). This
scene can be extended by considering the presence of another predator species
preying on the same prey. Models involving more than two species have been proposed
to describe some ecological phenomena, with very complex dynamical behaviors
exhibited (Naji and Balasim, 2007; Lv
and Zhao, 2008; Yu and Zhao, 2009; Upadhyay
and Naji, 2009).
In this study, the dynamical relations of two predator species predating on
a prey species were investigated. Various research approaches were undertaken
to analyze this particular species configurations (Dubey
and Upadhyay, 2004; Gakkhar et al., 2007).
The approach here was slightly taken different from others. From the persistency
conditions of the predators, the search and capture efficiency of the predators
was defined. The developed model and the methods of analysis were able to answer
some questions such as how the coexistence and extinction of the predators depend
on the efficiency of the search and the capture. Several numerical simulations
were carried out in the case of a non periodic solution.
MATHEMATICAL MODEL
According to Rockwood (2006), the diversity of organisms
and the difference in the environments have led to different models of population
growth. He added that the search for one model or one set of models for all
population in different environments is pointless. With this in mind, a new
model with the growth rates of the prey and two predators is described by the
logistic law, with the carrying capacity of the predators depending on the available
amount of prey is proposed. By using the Holling typeI functional response
to describe the predation of the two predators y and z on prey x, the model
can be written as:
The intrinsic growth rate of prey is r.α and β measure the efficiency
of search and capture of predators, z respectively. In the absence of prey x,
constants u and w are the death rates of predators y, z respectively. R_{1}
= e_{1}αx, R_{2} = e_{1}βx; R_{1}
and R_{2} represent numerical responses of the predators y, z, respectively
which describe change in the population of predators through prey consumption.
e_{1} and e_{2} represent efficiency of converting consumed
prey into predator births. The carrying capacities k_{x} = a_{1}x,
k_{z} = a_{2}z are proportional to the available amount of prey,
as was first proposed by Leslie (Gazi and Bandyopadhyay,
2008). c_{1} and c_{2} measure intraspecific competition
between the predators.
The system of Eq. 1 can be written in a nondimensional form.
This can be done in many ways but it is better that the choice of variables
relate to some key parameters. This makes the analysis less complicated as the
number of parameters is reduced from 12 to 8. Using the following transformations
of variables and parameters:
we have:
Equations 2 are of Kolgomorov type.
Theorem 1: The solution of the system (2) for t≥0 in
is bounded.
Proof: The first equation of the system (2) that represents the prey equation is bounded through:
The solution of the equation (3) is:
where:
is the constant of integration. Hence x(t)+y≤1, ∀t≥0. Next, we
prove thatx(t)+y(t)+z(t)≤L, ∀t≥0. Let D(t) = x(t)+y(t)+z(t). The
time derivative of the function D:
However, the solutions initiating remain in nonnegative quadrant in R^{3} and all the parameters are positive; it can be assumed the following:
It can be concluded that:
By substituting in Eq. 5 become as follows:
Equation 8 can be written as follows:
Since x(t)≤1, then:
But:
And:
Positive values are taken in (11) and (12).
So (10) becomes:
Where:
Consequently Z(t)≤L+σe^{1}, σ is a constant of integration.
As t→ ∞ we have D(t)≤L.
EQUILIBRIUM POINTS AND STABILITY ANALYSIS
Equilibrium points: It is observed that the system (2) has five nonnegative
equilibrium points. E_{0} = (0,0,0) and E_{1} = (0,0,0) are
obvious (i.e. they exist without conditions on parameters):
and
exist and are positive if the following conditions hold:
The fifth positive equilibrium point
will not be shown as it contains many parameters and hence will be very long.
Stability analysis: To study the local dynamical behavior of equilibrium points, the variational matrices of each equilibrium points are computed. From these matrices and using the RouthHurwitz criteria the local asymptotically stable are determined.
The variational matrix of E_{0} is given by:
It is observed from D_{0} that the manifold is unstable along xdirection but stable along ydirection and along zdirection because the eigenvalue of xdirection is positive, while the eigenvalues of ydirection and zdirection is negative. Therefore, the first equilibrium point E_{0} is saddle point.
The variational matrix of E_{1} is given by:
The equilibrium point E_{1} is locally asymptotically stable, provided the following conditions hold:
The variational matrix of E_{2} is given by:
From D_{2} and by using the RouthHurwitz criteria it is observed that the equilibrium point E_{2} is locally asymptotically stable, provided the following conditions hold:
The equilibrium point E_{2} is stable in xy plane if the condition (19) is satisfied. But the condition (15) must be satisfied so the condition (19) is always satisfied, while the stability of the E_{2} in the z direction (i.e. orthogonal direction to the xy plane) depend on the condition (20).
The variational matrix of E_{3} is:
The equilibrium point E_{3} is locally asymptotically stable with the following condition:
The equilibrium point E_{3} is stable in xz plane if the condition (21) is satisfied. On the other hand the condition (16) must be satisfied so the condition (21) is always satisfied, while the stability of the E_{3} in the y direction (i.e., orthogonal direction to the xz plane) depends on the condition (22).
For the equilibrium points:
the variational matrix is:
Where:
The characteristic equation of the variational matrix D_{4} is:
With:
From RouthHurwitz criterion:
is locally asymptotically stable if the following conditions hold:
We thus proved the following theorem.
Theorem (2):
• 
The equilibrium point E_{0} = (0,0,0) is a saddle
point with locally stable manifold in the yz plane and with locally unstable
manifold in the x direction 
• 
The positive equilibrium point E_{1} = (0,0,0) is locally asymptotically
stable in the xdirection but it is locally asymptotically stable in xz
plane if it holds the conditions (17) and (18). The equilibrium point E_{1}
is a saddle point if the conditions (17) and/or (18) are not satisfied 
• 
The equilibrium points: 
and:
• 
Are positive under the conditions (15) and (16) respectively.
The equilibrium point E_{2} is locally asymptotically stable provided
the conditions (19) and (20) hold, while the equilibrium point E_{3}
is locally asymptotically stable provided the conditions (21) and (22) hold 
• 
The nontrivial positive equilibrium point 
• 
Exists; it is locally asymptotically stable provided the conditions
(23) (24) and (25) hold 
Corollary: The equilibrium points E_{2} and E_{3} are unstable in zdirection (i.e. orthogonal direction to the xy plane) and in ydirection (i.e. orthogonal direction to the xz plane), respectively, if the condition (20) of E_{2} and the condition (22) of E_{3} are not satisfied (violated).
Theorem (3):
• 
The equilibrium point E_{2} is globally asymptotically
stable inside the positive quadrant of xy plane 
• 
The equilibrium point E_{3} is globally asymptotically
stable inside the positive quadrant of xz plane 
Proof: We prove (i) and in the same manner (ii) can be proved.
Let:
is a Dulac function, it is continuously differentiable in the positive quadrant of xy plane, A = {(x,y) x>0,y>0}.
Let:
Thus:
It is observed that Δ(GN_{1}, GN_{2}) is not identically
zero and does not change sign in the positive quadrant of xy plane A. So by
BendixsonDulac criterion (Logan and Wolesensky, 2009),
there is no periodic solution inside the positive quadrant of xy plane. E_{2}
is globally asymptotically stable inside the positive quadrant of xy plane.
Persistence and extinction: Freedman and Waltman
(1984) were studied system of equations of Kolgomorov type and derived abstract
theorems that showed persistency once certain conditions were applied. They
used a strong definition of persistency which is: A population x(t) is persistent
if x(0) > 0 and lim inft → ∞ x(t)>0. The system is said to
persist if each component of the system persists. The system (2) has no periodic
solution in the respective planes as was shown in Theorem 3. Also, the boundedness
of system (2) was proved in Theorem 1. According to Corollary 1, the orthogonal
directions of E_{2} and E_{3} are unstable if the conditions
(20) and (22) are not satisfied.
We next show that the conditions of Freedman and Waltman
(1984) are satisfied. We use y_{1} = y and y_{2} = z to
simplify the notations.
(C1) x is a prey population and y, z are competing predators, living exclusively
on the prey, i.e:
(C2) In the absence of predators, the prey species x grows to carrying capacity,
i.e.:
Here k = 1.
(C3) There are no equilibrium points on the y or z coordinate axes and no equilibrium point in yz plane.
(C4) The predator y and the predator z can survive on the prey, This means
that there exist points:
and:
such that:
and:
and .
Freedman and Waltman (1984) then showed that if the
above conditions hold, if there is no limit cycle and if:
then system (2) persists.
Inequalities (26) implies that:
Gakkhar et al. (2007) has mentioned the conditions
that represents the necessary conditions to include the following:
In the case of satisfying the conditions (27) and (28), then system (2) persists. However, in the case condition (27) was satisfied but condition (18) was not satisfied, then the first predator y survive, while the second predator z becomes extinct and vice versa.
Numerical simulations: Different values of the parameters α and
β in studying the dynamical behavior of the system numerically were considered.
The parameters α and β are important parameters because they are contained
in the functional and numerical responses which formed the main component of
prey predator models (Rockwood, 2006). In addition, they
are part of the intraspecific competition coefficients in our model. The functional
response plays an important role in interactions between prey and predator (Poggiale,
1998). α and β measure the efficiency of search and capture of
the predators. Two different cases were considered which showed the coexistence
and extinction of the predators. The other parameters were fixed in both cases.
However, the numerical simulations focused on showing the coexistence or extinction
of one of the predators. The values of parameters were chosen to satisfy the
stability conditions of equilibrium points E_{2} and E_{3} in
xy, xz planes, respectively which imply non periodic solution (Theorem 3).
The other parameters were fixed as follows:
Two different sets of numerical experiments were carried out. In the first
case, the value of β was fixed at 1.33 and the value of α varies.
It is observed in Fig. 1 that when the values of α and
β were near to each other, the three species stably coexist.

Fig. 1: 
Time series of dynamical behavior of the system (2) at α
= 1.37. Figure 1 described the coexistence of three species
(two predators one prey system) when α which is the value of efficiency
of search and capture of predator y was near of β that is efficiency
of search and capture of predator z, where the value of α was 1.37.
The dot dashed line represents prey x, the dashed line indicates predator
y, while the connected thick line represent predator z, this system of lines
is applied in all figures 

Fig. 2: 
Time series of dynamical behavior of the system (2) at α
= 0.9. Fig. 2 showed extinction of predator y and surviving
predator z when the value of efficiency of search and capture of predator
y (α) was decreased. Predator z is represented by the connected thick
line, predator y is indicated by the dashed line, while prey x is signified
by the dot dashed line 

Fig. 3: 
Time series of dynamical behavior of the system (2) at α
= 1.9. Figure 3, extinction of predator z and surviving
predator y was shown when increasing the value of efficiency of search and
capture of predator y (α) to become 1.9. The dashed line indicates
predator y, predator z is represented by the connected thick line, while
the dot dashed line represents prey x 
However, if the value of α was decreased (α = 0.9), predator y
became extinct, while predator z survived, as is shown in Fig.
2. On the other hand, if α was increased (α = 1.9), the predator
z tended to extinction (Fig. 3) and predator y survived. This
showed that the survival of each predator depended on the efficiency of the
search and capture.
In the second case, the same values of parameters were used but the value of
α was fixed at 1.1 and giving different values to β.

Fig. 4: 
Time series of dynamical behavior of the system (2) at β
= 1.3. In Fig. 4, the coexistence of three species (two
predators one prey system) is shown when β that represents the value
of efficiency of search and capture of predator z is near of value β
which is efficiency of search and capture of predator y, the value of β
is 1.3. The dashed line represents predator y, the connected thick line
represent predator z, while the dot dashed line represents prey x 

Fig. 5: 
Time series of dynamical behavior of the system (2) at β
= 0.9. Figure 5 illustrates extinction of predator z and
surviving predator y when decreasing the value of efficiency of search and
capture of predator z (β) to become 0.9. The dashed line indicates
of predator y, predator z is represented by the connected thick line, while
the dot dashed line represents prey x 
Corresponding results for β are shown in Fig. 4, Fig.
5 and Fig. 6. when the values of α and β were
almost near to each other where three species may coexist at these values as
it is shown in Fig. 4. But in case the value of β(β
= 0.9) was decreased, predator z became extinct, while predator y survived,
as is clear in Fig. 5. Otherwise if β was increased (β
= 1.8), the predator y tends to extinction and predator z survives, this is
explained in Fig. 6.

Fig. 6: 
Time series of dynamical behavior of the system (3) at β
= 1.8. Figure 6 shows surviving predator z and extinction
of predator y when increasing the value of β to become 1.8. The connected
thick line represents predator z, the dashed line indicates of predator
y, while the dot dashed line represents prey x 
It was observed that the numerical simulations correspond with theoretical
analysis when the conditions in section 5 were applied.
Predators and prey can influence one another's evolution. Traits that enhance
a predator's ability to find and capture a prey will be selected in the predator.
Traits that enhance a prey's ability to avoid being captured and eaten by a
predator will be selected in the prey. The “goals” of these traits
are not compatible and it is the interaction of these selective pressures that
influences the dynamics of the predator and prey populations (ElMessoussi
et al., 2007; Younas et al., 2004;
Islam et al., 2004). Predicting the outcome of
species interactions is also of interest to biologists trying to understand
how communities are structured and sustained. One aspect that may affect the
efficiency of searching and capture by predators is the existing environmental
effects interfere with their foraging activities (Myers
et al., 2007; Smee, 2010). Recent research
has shown that the movement properties of foraging animals may have important
implications for their success in locating prey (Scharf
et al., 2006).
CONCLUSIONS
In this study a continuous time mathematical model of interactions of two competing
predators sharing one prey was introduced. Holling typeI functional responses
have been used. The conditions of existence of equilibrium points and their
stability of equilibrium points of the model were obtained. Theoretical analysis
on persistence of the system and the extinction of one of the predators was
presented.
Numerical simulations showed that if the efficiency of searching and capture of both predators was roughly the same, the three species can coexist. However, if the efficiency of one of the predator was less than the other, this leaded to the extinction of the later predator.
ACKNOWLEDGMENTS
This research is supported by a grant from the School of Mathematical Sciences, Universiti Sains Malaysia.