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# The Influence of Intra-specific Competition Among Predator Populations

- Subject:
**Business**,**Life**,**Sociology** - Category:
**Strategy**,**Professions & Career**,**Anthropology** - Topic:
**Competition**,**Model**,**Population** - Pages 4
- Words: 1937
- Published: 12 Jun 2019
- Downloads: 46

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Abstract

The present investigation accounts for the influence of intra-specific competition among predator populations in a prey dependent predator-prey model of interacting populations. In this model we considered Qiwu’s growth rate for prey, Crowley-Martin response function for predation and intra-specific competition among predator populations. We offer a detailed mathematical analysis of the proposed model to illustrate some of the significant results that has arisen from the interplay of deterministic ecological phenomena and processes. In particular, stability (local and global) and bifurcation ( Hopf-Andronov) analysis of this model are conducted for biologically feasible parameters. We attained sufficient conditions to ensure the global stability of the unique positive equilibrium, by using appropriate Lyapunov functions and Lasalle invariance principle. Finally, numerical simulations through computer illustrate the dynamics of the ecological system.

1. Introduction

Ecological systems are characterized by the interactions of different species within their natural environment. Studies of mathematical models are informative in understanding predator-prey interactions in these systems; as a result, predator-prey models have been important in ecological science since the early days of this discipline after the pioneering works of Lotka and Volterra. We start with the modified Lotka-Voltera type predator-prey system

dX/dT=RX(1-X/K)-CMXY, (1.1a)

dY/dT=MXY-DY, (1.1b)

X(0)≥0,Y(0)≥0, (1.1c)

Where X(t) and Y(t) are the population densities of prey and predator respectively; R is the logistic growth rate, K is the environmental carrying capacity of prey, C is the conversion factor. M is the predation rate of predator and D is the death rate of predator.

Qiwu and Lawson (1982) proposed a new equation concerning the growth of a single species population using absorption theory of chemical kinetics and conclude that the growth of a single species population in nature seems to support the new equation rather than the logistic equation, and it can be successfully fitted by means of a least square method. But most of the researchers develop prey-predator model considering logistic growth proposed by Verhulst till now (cf. (Haque, 2009; Abrams, 1994; Jost & Arditi, 2000; Freedman, 1980; Ko & Ryu, 2006; Haque, Rahaman, Venturino & Li, 2014)).

Crowley and Martin proposed a functional that can accommodate interference among predators (see Crowley, P.H., Martin, F. K. (1989) Functional responses and interference within and between year classes of a dragonfly population, Journal of the North American Benthological Society, 8(3) : 211-221). This type of functional response is classified as one of predator-dependent functional response, i.e. they are functions of the abundance of both prey and predator due to predator interference. It is assumed that predator-feeling rate decreases by higher predator density even when prey density is high, and therefore the effects of predator interference on feeding rate remain important all the time whether an individual predator is handing or searching for a prey at a given instant of time. The per capita feeding rate in this formulation is given by

p(u,v)=Mu/(1+Au+Bv+ABuv)

Where M, A and B are positive parameters that describe the effects of capture rate, handling time and the magnitude of interference among predators, respectively, on the feeding rate.

The objective of this paper is to analyze prey-predator system with Qiwu’s growth function in replace of logistic growth function in a classical prey-predator model. We consider Crowley Martin response function in presence of intra-specific competition among predators.

The rest of this article is organized as follows. In Section 2, we present our mathematical model with Crowley Martin response function for predation. Some preliminary results such as existence and boundedness of the system solution in Section 3. Section 4 presents the behaviour of the system around all positive equilibrium points including local stability, global stability and bifurcation. Direction and stability of periodic orbit generated through hopf-bifurcation is discussed in Section 5. Section 6 supports analytical findings through numerical simulation. The article concludes with a discussion in Section 7.

2. The model:

According to the above assumptions, the model (1.1) becomes

dX/dT=RX(G_0-X)/((G_1-X) )-CMXY/(1+AX+BY+ABXY) , (2.1a)

dY/dT=MXY/(1+AX+BY+ABXY)-DY-HY^2 , (2.1b)

X(0)≥0,Y(0)≥0, (2.1c)

where the system parameters G_0,G_(1,) M,C,A,D,H are all positive constants; M,C,D are defined earlier in model (1.1); 〖RG〗_0 is the carrying capacity of the prey species, G_1 is the value of limiting resources. In other words, it is theoretic carrying capacity if there is no wastage in resources under ideal conditions, which is impossible in reality. G_0/G_1 concerns the efficiency of nutrient utilization by a species. Its value is between zero and one. With ratios approaching unity, the efficiency is high; lower ratios indicate that population increment is quickly restricted by limiting resources (Qiwu and Lawson,1982); H is the co-efficient of intra-specific competition among predators.

Using the transformations x=X/R,y=YC/R,t=DT, the system (2.1) takes the following form

dx/dt=rx(k_0-x)/((k_1-x) )-mxy/(1+ax+by+abxy) , (2.2a)

dy/dt= mxy/(1+ax+by+abxy)-y-hy^2 , (2.2b)

X(0)≥0,Y(0)≥0 (2.2c)

where h=HR/CD,r=R/D,k_0=G_0/R,k_1=G_1/R,m=MR/D,a=AR,b=BR/C .

3. Preliminary results

3.1. Existence and positive invariance

For t>0, letting X≡〖(x,y)〗^(T ),F:R^2→R^2 ,F=(f_1,f_2 )^T, system (2.2) can be written as dX/dT=F(X). Here f_i∈C^∞ (R) for i=1,2; where f_1=rx(k_0-x)/((k_1-x) )-mxy/(1+ax+by+abxy) ,f_2=mxy/(1+ax+by+abxy)-y-hy^2 .

Since the vector function F is a smooth function of the variables (x,y) in the positive quadrant

Ω={(x,y): x>0,y>0}, the local existence of the solution hold.

3.2. Boundedness results

Boundedness implies that the system is biologically well behaved. The following propositions ensure the boundedness of the system (2.2).

3.2.1. The prey population is always bounded from above.

Proof. From the first equation of the system (2.2) it can be shown that

lim┬(n→∞)〖Sup x(t)≤k_0 〗

3.2.2. The predator population is bounded.

Proof. From the second equation of the system (2.2)

dy/dt= mxy/(1+ax+by+abxy)-y-hy^2≤mxy/(1+ax)(1+by) -y≤(mk_0 y)/(1+ak_0 )-y=((mk_0)/(1+ak_0 )-1)y

⇒y(t)≤y(0)exp{((mk_0)/(1+ak_0 )-1)t}

which is finite for t<∞. Hence the claim.

3.3.Equilibria and their feasibility

System (2.2) has the following three positive equilibria (2.2) E_i (x_i,y_i ),i=1,2,3.E_0 is the origin, E_(1 )≡(k_0,0).

For E_2 (x_2,y_2), we have y_2=r(k_0-x_2 )(1+ax_2 )/(m(k_1-x_2 )-rb(k_0-x_2 )(1+ax_2 ) )

and x_2 is a positive root of the equation

B_5 x^5+B_4 x^4+B_3 x^3+B_2 x^2+B_1 x+B_0=0,

where

B_5=-a^2 b^2 r^2 m,

B_4=-a^2 bmr+2abm^2 r+2a^2 b^2 k_0 mr^2-2ab^2 mr^2+a^2 hmr

B_3=-2abmr+b^2 r^2 (-2a^2 k_0+2a)+(a-m)[m^2-2bmr(-ak_0-ak_1+1)+b^2 r^2 (a^2 〖k_0〗^2+1-4ak_0 ) ]+mr(b+h)(-a^2 k_0-a^2 k_1+2a)-b^2 r^2 (a^3 〖k_0〗^2-6a^2 k_0+3a)

B_2=m^2-2bmr(-ak_0-ak_1+1)+b^2 r^2 (a^2 〖k_0〗^2+1-4ak_0 )+(a-m)[-2k_1 m^2-2bmr(ak_0 k_1-k_0-k_1 )+b^2 r^2 (2a〖k_0〗^2-2k_0 ) ]+mr(b+h)(a^2 k_0 k_1-2ak_0-2ak_1+1)-b^2 r^2 (3a^2 〖k_0〗^2-6ak_0+1)

B_1=-2k_1 m^2-2bmr(ak_0 k_1-k_0-k_1 )+b^2 r^2 (2a〖k_0〗^2-2k_0 )+(a-m)[m^2 〖k_1〗^2-2bmrk_0 k_1+b^2 〖〖k_o〗^2 r〗^2 ]+mr(b+h)(2ak_0 k_1-k_0-k_1 )-b^2 r^2 (3a〖k_0〗^2-2k_0 )

B_0=m^2 〖k_1〗^2-bmrk_0 k_1+hk_0 k_1 mr.

4. Stability analysis

The Jacobian matrix of the system (2.2) at any arbitrary point is given by

J(x,y)=[■((r(k_0 k_1-2k_1 x+x^2))/(k_1-x)^2 -my/((1+by) (1+ax)^2 )&-mx/((1+ax) (1+by)^2 )@my/((1+by) (1+ax)^2 )&mx/((1+ax) (1+by)^2 )-1-2hy)] (4.1)

4.1. System behaviour near the origin

The Jacobian matrix at the equilibrium E_0 is J_0=[■((rk_0)/k_1 &[email protected]&-1)].

The eigenvalues of the Jacobian matrix J_0 at E_0 are (rk_0)/k_1 ,-1. Hence E_0 is unstable in nature.

4.2. System behaviour near the equilibrium E_(1 ) (k_0,0)

(i) E_(1 )is locally asymptotically stable if 00,y≥0} and consider the scalar function L_1:R^2→R defined by

L_1=1/2 (x-k_0 )^2+k_0 y (4.2)

(dL_1)/dt=(x-k_0 ) dx/dt+k_0 dy/dt=(x-k_0 )[rx(k_0-x)/((k_1-x) )-mxy/(1+ax)(1+by) ]+k_0 [mxy/(1+ax)(1+by) -y-hy^2 ]

=-(rx(k_0-x)^2)/((k_1-x) )-mxy(〖x-k〗_0 )/(1+ax)(1+by) +(k_0 mxy)/(1+ax)(1+by) -k_0 y-hk_0 y^2

=-(rx(k_0-x)^2)/((k_1-x) )-(mx^2 y)/(1+ax)(1+by) +(2k_0 mxy)/(1+ax)(1+by) -k_0 y-hk_0 y^2

≤-(rx(k_0-x)^2)/((k_1-x) )-mxy(〖x-k〗_0 )/(1+ax)(1+by) +2〖k_0〗^2 my-k_0 y-hk_0 y^2

≤0 if k_0<1/2m (4.3)

and (dL_1)/dt=0 when (x,y)=(k_0,0). The proof follows from (4.3) and Lyapunov-Lasale invariance principle (Hale, 1969).

4.3. System behaviour near the equilibrium E_2 (x_2,y_2 )

(i) E_(2 )is locally asymptotically stable if y_2<(rx_2 (k_1-k_0 ) (1+ax_2 )^2)/(x_2 [am(k_1-x_2 )^2-br(k_1-k_0 ) (1+ax_2 )^2 ] ) and k_00,c_22=-(bmx_2 y_2)/((1+ax_2 ) (1+by_2 )^2 )-hy_2<0. Its eigenvalues are

λ_1,2=1/2 [(c_11+c_22 )±√((c_11+c_22 )^2-4(c_11 c_22-c_12 c_21 ) )]. (4.4)

If we assume c_11<0 then λ_1,2both are either negative or complex numbers with negative real parts. Hence, E_(2 )is locally asymptotically stable if c_11<0 that is, y_2<(rx_2 (k_1-k_0 ) (1+ax_2 )^2)/(x_2 [am(k_1-x_2 )^2-br(k_1-k_0 ) (1+ax_2 )^2 ] ) and

k_00,y>0} and consider the scalar function L_1:R^2→R defined by

L_2={(x-x_2 )-x_2 ln x/x_2 }+P{(y-y_2 )-y_2 ln y/y_2 }, (4.5)

where P is a positive constant determined latter. The derivative of the above equation (4.6) along the solution of the system (4.2) is given by

(dL_2)/dt=(1-x_2/x) dx/dt+P(1-y_2/y) dy/dt=(1-x_2/x)[rx(k_0-x)/((k_1-x) )-mxy/(1+ax)(1+by) ]+P(1-y_2/y)[mxy/(1+ax)(1+by) -y-hy^2 ]= (x-x_2 )[r(k_0-x)/((k_1-x) )-my/(1+ax)(1+by) ]+P(y-y_2 )[mx/(1+ax)(1+by) -1-hy] (4.6)

At the equilibrium point E_(2 )of the system (4.2), we have

r(k_0-x_2 )/((k_1-x_2 ) )-(my_2)/(1+ax_2 )(1+by_2 ) =0, (mx_2)/(1+ax_2 )(1+by_2 ) -1-hy_2=0 (4.7)

Using (4.7) the time derivative of L_2 becomes

(dL_2)/dt= (x-x_2 )[r(k_0-x)/((k_1-x) )-r(k_0-x_2 )/((k_1-x_2 ) )+(my_2)/(1+ax_2 )(1+by_2 ) -my/(1+ax)(1+by) ]+P(y-y_2 )[mx/(1+ax)(1+by) -(mx_2)/(1+ax_2 )(1+by_2 ) +hy_2-hy]

=(x-x_2 )^2 [-(r(k_1-k_0))/(k_1-x)(k_1-x_2 ) +(amy_2)/(1+ax)(1+ax_2 )(1+by_2 ) ]+(x-x_2 )(y-y_2 )[(Pm-m-amx_2+bmPy_2)/(1+ax)(1+by)(1+ax_2 )(1+by_2 ) ]+(y-y_2 )^2 [-hp-(bmpx_2)/(1+by)(1+ax_2 )(1+by_2 ) ]

=(x-x_2 )^2 [-(r(k_1-k_0))/(k_1-x)(k_1-x_2 ) +(amy_2)/(1+ax)(1+ax_2 )(1+by_2 ) ]-(y-y_2 )^2 [hp+(bmpx_2)/(1+by)(1+ax_2 )(1+by_2 ) ] [Taking Pm-m-amx_2+bmPy_2=0,that is P=((1+ax_2 ))/((1+by_2 ) )>0]

≤(x-x_2 )^2 [-(r(k_1-k_0))/(k_1-x)(k_1-x_2 ) +(amy_2)/(1+ax_2 )(1+by_2 ) ]-(y-y_2 )^2 [hp+(bmpx_2)/(1+by)(1+ax_2 )(1+by_2 ) ]

≤-[a_11 (x-x_2 )^2+a_22 (y-y_2 )^2]

≤0 if a_11>0, (4.6)

where a_11=(r(k_1-k_0))/(k_1-x)(k_1-x_2 ) -(amy_2)/(1+ax_2 )(1+by_2 ) ,a_22=hp+(bmpx_2)/(1+by)(1+ax_2 )(1+by_2 ) ,

and (dL_2)/dt=0 when (x,y)=(x_2,y_2 ). The proof follows from (4.6) and Lyapunov-Lasale invariance principle (Hale, 1969).

5. Stability and direction of Hopf-bifurcation

In this section, our attention is focused on investigation of the stability of the periodic solution bifurcating from a stable equilibrium E_2 (x_2,y_2 ). For this, we consider u=x-x_2,v=y-y_2. Then the system (2.2) reduces to

du/dt=b_1 u+b_2 v+∑▒〖b_ij u^i v^j 〗 ,

dv/dt=c_1 u+c_2 v+∑▒〖c_ij u^i v^j 〗

i≥0,j≥0,i+j≥2,

Where

b_1=(r(k_0 k_1-2k_1 x_2+〖x_2〗^2))/(k_1-x_2 )^2 -(my_2)/((1+by_2 ) (1+ax_2 )^2 ) , b_2=-(mx_2)/((1+ax_2 ) (1+by_2 )^2 ),

〖c_1=(my_2)/((1+by_2 ) (1+ax_2 )^2 ),c〗_2=(mx_2)/((1+ax_2 ) (1+by_2 )^2 )-1-2hy_2-〖y_2〗^2

b_20=(rk_1 (k_0-k_1))/(k_1-x_2 )^3 +(amy_2)/((1+by_2 ) (1+ax_2 )^3 ),b_11= -(mx_2)/((1+ax_2 )^2 (1+by_2 )^2 ),b_02=(bmx_2)/((1+ax_2 ) (1+by_2 )^3 ),

c_20=-(amy_2)/((1+by_2 ) (1+ax_2 )^3 ),c_11=m/(1+by_2 )(1+ax_2 ) ,c_02=-bm/((1+ax_2 ) (1+by_2 )^3 )-h,

b_30=(rk_1 (k_0-k_1))/(k_1-x_2 )^4 -(a^2 my_2)/((1+by_2 ) (1+ax_2 )^4 ),b_21=am/((1+ax_2 )^3 (1+by_2 )^2 ),b_12=bm/((1+ax_2 )^2 (1+by_2 )^3 ),b_03=-(b^2 mx_2)/(1+by_2 )^3 ,

c_30=(a^2 my_2)/((1+by_2 ) (1+ax_2 )^4 ),c_21=-am/((1+ax_2 )^3 (1+by_2 )^2 ),c_12=-bm/((1+ax_2 )^2 (1+by_2 )^3 ),c_03=(b^2 mx_2)/(1+by_2 )^3 .

Following the book of (Perko, 2006), one can obtain the Liyapunov number σ as follows:

σ=-3π/(2b_2 ∆^(3⁄2) ) [{b_1 c_1 (〖b_11〗^2+b_11 c_02+b_02 c_11 )+b_1 b_2 (〖c_11〗^2+b_20 c_11+b_11 c_02 )+〖c_1〗^2 (b_11 b_02+2b_02 c_02 )-2b_1 c_1 (〖c_02〗^2-b_20 b_02 )-2b_1 b_2 (〖b_20〗^2-c_20 c_02 )-〖b_2〗^2 (2b_20 c_20+c_11 c_20 )+(b_2 c_1-2〖b_1〗^2 )(c_11 c_02-b_11 b_20 ) }-(〖b_1〗^2+b_2 c_1 ){3(c_1 b_03-b_2 b_30 )+2b_1 (b_21+c_12 )+(c_1 b_12-b_2 c_21 ) } ] (5.2)

The bifurcating periodic solutions are stable (unstable) if σ>0(σ<0). Hence, Hopf-bifurcation is supercritical (subcritical) if >0(σ<0) .

6. Numerical analysis

Theoretical analysis cannot be completed without numerical validation. In this article, we perform numerical simulation of the system (2.2) using Runge-Kutta 4th order method by Matlab R2010a and Mapple 18. In Figure 1, global stability around 〖 E〗_1 of the system (2.2) is shown. The system (2.2) can be locally asymptotically stable if k_0=1.0<〖k_0〗^[HB] =1.566123226 and the system experiences Hopf bifurcation around 〖 E〗_2 if k_0=2.2>〖k_0〗^[HB] =1.566123226 . As the value of Liyapunov number σ=4946.655695>0, the bifurcating periodic orbits are stable and the Hopf bifurcation is supercritical. Oscillatory behaviour of the system (2.2) becomes asymptotically stable in presence of intra specific competition among predator populations.

7. Discussion and conclusion

In this work, a bidimensional continuous-time differential equations system has analysed which

is derived from a Logistic-type predator–prey model by considering a Crowley Martin functional response. We made a reparameterization and a time rescaling to obtain a topologically equivalent system in order to facilitate calculus. Although bifurcations in a predator-prey model with Verhulst’s logistic growth rate have been studied by many researchers (Fan and Kuang, 2004), (Xiao and Ruan, 2001), (Hwang, 2003), (Bohner, Fan and Zhang, 2006), there is no article on the stability and bifurcation properties of predator-prey model with Qiwu’s growth rate for prey and Crowley-Martin functional response for predation process. For this model, We have dealt with local and global stability, stability of Hopf-bifurcation in an orderly manner. A complete classification of the equilibrium points, with respect to the various parameters based on their existence conditions, is provided. We have shown in Section 4 that the local and global stability properties of the system (2.2) around respective equilibrium points and in Section 5 that the system experiences the Hopf-bifurcation as k0 crosses its critical value 〖k_0〗^[HB] . The normal form theory and center manifold reduction have been made use of and we have derived the explicit formula which determine the stability property of bifurcating periodic solutions. At the end, it can be concluded that the observations that have been made in this article will help the experimental setups, and as a result, development of theoretical ecology will be enhanced in some measure in the near future.

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