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In this paper, we offer and investigate stability (local and global) and bifurcation (Hopf-Andronov) analyses of a Crowley-Martin predator-prey model system with Qiwu’s growth for prey species and intra-specific competition among predator species. Initially, by virtue of the comparison principle, we achieve boundedness results for both the interacting species. Subsequently, by means of linear stability analysis of the proposed model, we demonstrate that under some parametric restrictions the unique positive equilibrium solution is locally as well as globally stable. Consequently, the direction of Hopf bifurcation and stability of bifurcating periodic solutions are studied. At the end of the investigation, we wind up our findings via intra-specific competition among predator species and present a concise discussion.

Ecosystems are differentiated by the interaction of species contained by their natural atmosphere. Explorations of ecological models via mathematical analysis are useful in understanding predator-prey interactions that influence the dynamics of all species; accordingly, predator-prey models have been significant as well as one of the core research subject in ecological science since the pioneering work of Lotka and Volterra. We initiate with the basic Lotka-Volterra predator-prey model with logistic growth in prey species

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 densities of prey and predator species respectively. The parameters emerging in the equation (1.1) are clarified below

R: the prey’s intrinsic growth rate in the absence of predation,

K: the environmental carrying capacity of prey species,

C: the conversion factor,

M: the rate at which prey is foraged and it follows Holling type-I functional response,

D: the mortality rate of predator species.

In 1982, Cui Qiwu and G. J. Lawson (1982) offered another type of population growth with reference to the growth of a single species by means of absorption theory of chemical kinetics and wind up that the growth of a single species biologically looks to carry the revised equation more willingly than the logistic equation, and it can be effectively fixed through a least square method. Although most of the research scholars build up predator-prey system taking into accounts the Verhulst’s logistic growth so far (cf. (Haque, 2009; Abrams, 1994; Jost & Arditi, 2000; Freedman, 1980; Ko & Ryu, 2006; Haque, Rahaman, Venturino & Li, 2014)).

The Crowley-Martin type of functional response was initiated by Crowley and Martin that can put up interference among predator species (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). It is well known that Crowley-Martin type functional response is fundamentally predator-dependent functional response, that is, they are functions of the large quantity of both the interacting species owing to predator interference. It is believed that predator-feeding rate reduces by higher predator size still when prey size is high, and consequently the results of predator interference on feeding rate continuously essential constantly whether an individual predator species is passing or penetrating for a prey species at a particular moment. The per capita feeding rate in this formulation is specified by

f(X,Y)=MX/(1+AX+BY+ABXY)

where M, A, B are positive constants that illustrate the effects of capture rate, handling time and the magnitude of interference among predators, respectively, on the feeding rate.

The main purpose of this investigation is to explore predator-prey model system with prey’s Qiwu’s growth function instead of usual modified Lotka-Volterra (that is logistic growth for prey species) predator-prey model. We think about dynamical behaviour of the proposed model with Crowley-Martin type predation and intra-specific competition among predator species.

The rest of the paper is organized as follows. The Crowley-Martin predator-prey model with Qiwu’s growth for prey is introduced in Section 2. In Section 3, we offer some preliminary results such as existence and boundedness of the proposed model system. In Section 4, we provide linear stability analysis of uniform feasible situation for the system via Routh-Hurwitz criterion. Direction and stability of periodic orbit created through Hopf-bifurcation is talked about in Section 5. Section 6 sustains analytical result via numerical simulation. The paper finishes with a discussion in Section 7.

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.

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 .

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 .

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

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

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

lim┬(n→∞)〖Sup x(t)≤k_0 〗

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.

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.

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)

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.

(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).

(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).

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) .

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. Figure 2 shows that 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. Global stability around interior equilibrium position 〖 E〗_2 of the system (2.2) is depicted in Figure 3. Figure 4 shows that oscillatory behaviour of the system (2.2) becomes asymptotically stable in presence of intra specific competition among predator populations.

In this rigorous investigation, a two-dimensional (2D) continuous-time differential equations system has analyzed which is originated from a Crowley Martin predator-prey model by considering Qiwu’s growth function for prey species. We prepared a reparameterization and a time rescaling to acquire a topologically equivalent system in order to make easy calculus.

Even though bifurcations in a predator-prey model with Verhulst’s logistic growth function for prey species have been considered by many distinguished researchers (Fan and Kuang, 2004), (Xiao and Ruan, 2001), (Hwang, 2003), (Bohner, Fan and Zhang, 2006), there is no expose on the stability and bifurcation properties of a Crowley-Martin predator-prey model with Qiwu’s growth rate for prey species.

For the proposed model, we have dealt with local and global stability, stability of Hopf-bifurcation in a methodical approach. A complete classification of the fixed points, with respect to the diverse system parameters based on their existence conditions, is provided. We have revealed 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 cross its critical value〖k_0〗^[HB] . The normal form theory and center manifold reduction have been prepared use of and we have developed the explicit formula which resolve the stability property of bifurcating periodic solutions. In the last part, it can be concluded that the examinations that have been made in this critique will help the experimental setups, and consequently, development of theoretical as well as mathematical ecology will be developed in some measure in the near future.

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