DYNAMICS OF A RATIO-DEPENDENT PREY-PREDATOR SYSTEM WITH SELECTIVE HARVESTING OF PREDATOR SPECIES. 1. Introduction

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1 J. Appl. Math. & Computing Vol ), No. 1-2, pp Website: DYNAMICS OF A RATIO-DEPENDENT PREY-PREDATOR SYSTEM WITH SELECTIVE HARVESTING OF PREDATOR SPECIES TAPAN KUMAR KAR Abstract. The dynamics of a prey-predator system, where predator population has two stages, juvenile and adult with harvesting are modelled by a system of delay differential equation. Our analysis shows that, both the delay and harvesting effort may play a significant role on the stability of the system. Numerical simulations are given to illustrate the results. AMS Mathematics Subject Classification : 34K20, 92 D25. Key words and phrases : Ratio-dependent, delay, harvesting, limit cycles, Hopf-bifurcation. 1. Introduction Renewable resources management is complicated and constructing accurate mathematical models about the effect of harvesting is even more complicated. This is so because to have a perfect model we would have to take into account many factors on the survival of the harvested population. For each population we need to consider its size, growth rate, carrying capacity, predators, competitors, environmental fluctuation, etc. But it is obvious that a perfect model cannot be achieved because even if we could put all these factors in a model, the model could never predict ecological catastrophes or Mother Nature caprice. Therefore, the best we can do is to look for analyzable models that describe as well as possible the reality or the effect of harvesting on populations. The study of population dynamics with harvesting is a very interesting subject in mathematical bioeconomics. It is related to the optimal management of renewable resources [4]. The exploitation of biological resources and the harvest of population species are commonly practised in fishery management. The basic models are usually in the form of differential equations. Received May 5, c 2007 Korean Society for Computational & Applied Mathematics and Korean SIGCAM. 385

2 386 Tapan Kumar Kar Brauer and Soudack in [2] discussed the following model ẋ = rx 1 x ) yx k a + x, ẏ = y d + x ) h 1) a + x where k is the carrying capacity of the prey population, d is the death rate of the predator, r is the intrinsic growth rate of the prey population and h is x the constant harvesting rate. The function is often called the functional a + x response of Holling type II. k,d,r,aand h are positive constants. An alternative assumption of the model 1) is that as the numbers of predator change slowly relative to prey change), there is often competition among the predators and the per capita rate of predator depends on the numbers of both prey and predator, most likely and simply on their ratio. Generally, a ratiodependent predator-prey model takes the form x ) x ) ẋ = xfx) yp, ẏ = cyp dy 2) y y Particularly for the ratio-dependent type predator-prey model with Michaelis- Menten type functional response, the system 2) takes the form ẋ = ax 1 x ) cxy k my + x, ẏ = y d + fx ), 3) my + x k is the carrying capacity of the prey, and a, c, m and f are positive constants that stand for prey intrinsic growth rate, capturing rate, half saturation constant and conversion rate, respectively. Motivated by recent works of Kuang and Beretta [19], and Kuang [20], Fan and Kuang [8], Fan et al. [9], Fan and Wang [10, 11] we consider the ratio dependent model rather than the model system 1) as done by Brauer and Soudack [2]. Here we consider that, y represents the adult predator. The assumption in this model is that consumption of prey is translated instantaneously into adult predators. However, there is an interpretation of this model that makes reasonable biological sense. Suppose d = d o b o, where b o is a constant birth rate, and d o is the maximum death rate in the absent of food d o <b o ). Consumption of prey by the predator then acts to stave off starvation by decreasing d o. Thus the predator equation can be written in the form { ẏ = b o y d o fx } y my + x This assumes that all the metabolic energy a predator obtains from its food goes into physiological maintenance, rather than into enhancing reproductive effort. Next we assume that newborns take τ units of time to become mature. Under this assumption model becomes ) ẋ = ax cxy 1 x k ẏ = pb o yt τ) my + x { d o fx my + x } y, 4)

3 Dynamics of a ratio-dependent prey-predator system 387 where b o yt τ) is the rate of production of newborns at time t τ, and p is the fraction of these that survive to maturity at time t. Here for simplicity we take p =1. Note that this model assumes that only adults feed on the prey x: juveniles are assumed to have some independent diet. This is reasonable for those species that need to grow sufficiently large before they can catch their prey. On logical consideration, random fishing of all fishes is not advisable for the persistence of the fishery. Generally speaking, the exploitation of population should be from the mature population which is more appropriate to the economic and biological views regarding renewable resources management. From this basic stand point we assume that only mature predators are subjected to harvesting. Thus our model becomes ) ẋ = ax cxy 1 x k ẏ = pb o yt τ) my + x, { d o fx my + x } y h, 5) where the harvesting term h = qey is based on the catch-per-unit effort hypothesis [4]. Here q is the catchability co-efficient, E is the harvesting effort. This type of harvesting can be made by adjusting the mesh size of the net so that when nets are placed in water, captured all fish except those that are small enough to swim through the water. The initial conditions are x0) = x o > 0 and yθ) =φθ) 0,θ [ τ,0]. Good reviews are given in Cushing [5], MacDonald [21], Gopalswamy [13], Driver [7], Mukhopadhyay et al. [22] and Kar [15-18], Song and Chen [24] Chattopadhyay et al. [3], Samanta et al. [23], Zhang et al. [25] and the references cited therein, regarding time delay and harvesting models. 2. Dynamics of the system System 5) always has equilibria 0, 0), k, 0) and has unique positive equilibrium P x,y ) if and only if one of the following two conditions are true: i) 1 [ ] f c ma) d <E< 1 f d) when c > ma, q c q ii) E< 1 f d), when c ma. q In both cases, we have x = cd + qe) fc ma) K, y = x f d qe) amf d + qe)m. We assume that the system parameters satisfy the above condition. We see that x increase with E, which is natural as an increase in E decreases the predator population and hence the survival rate of the prey.

4 388 Tapan Kumar Kar MSY he) E MSY E Figure 1. The figures shows that he) increases for E<5.1, decreases for E>5.1 and MSY occurs at E =5.1 [ ] Again we observe that if c>ma,y increases for E < 1 c ma f d, q c [ ] decreases for E > 1 c ma f d and attains maximum value at E = q c [ ] 1 c ma f d. If c ma, then y decreases with E. q c Maximum sustainable yield MSY) of a biological resource population is the maximum rate at which it can be harvested even after maintaining the population at a constant level. Therefore, corresponding to a given effort E, the sustainable yeild he) is given by he) =qey = kqe am 2 d + qe)f [ ] cd + qe)f d qe) fc am)f d qe). Analytically it is very difficult to find MSY and corresponding E MSY. For simulation we take the parameter values as a = 1.0, c = 0.9, k = 400, m = 1, d= 0.07, f = 0.4, q= For these values of parameters it is found that MSY =29.0 and corresponding E MSY =5.1. For any value of E>E MSY, the yield he) monotonically decreases with E towards zero see Figure 1), Biologists call it a case of biological over exploitations. We have concentrated mainly on the interior equilibrium of the system as we are interested in the coexistence of the species. Let X = x x,y = y y are the perturbed variables. After removing non-linear terms, we obtain the linear

5 Dynamics of a ratio-dependent prey-predator system 389 variable system, by using equilibrium conditions as dx a dt = cx y ) k x my + x ) 2 X dy [b dt = o + mfx y ] my + x ) 2 Y + cx 2 my + x ) 2 Y mfy 2 my + x ) 2 X + b oy t τ) 6) From the linearised system we obtain the characteristic equation where, F λ, τ) =λ 2 + Aλ +Bλ + F )e λτ + C = 0 7) A = a cx y k x my + x ) 2 + b o + mfx y my + x ) 2 B = b o a cx y ) F = b o k x my + x ) 2 C = mfcx 2 y 2 a my + x ) 4 + cx y k x )b my + x ) 2 o + mfx y Case I. τ =0. For τ = 0, the characteristic equation 7) becomes my + x ) 2 ). λ 2 +A + B)λ + C + F = 0 8) Now, sum of the roots = A+B) and product of the roots = C +F. Therefore, we can say that both the roots of 8) are real and negative or complex conjugate with negative real parts if and only if A + B>0 and C + F>0. After some algebraic manipulation it can be shown that the above conditions are satisfied when mf > c. Hence P x,y ) is locally asymptotically stable if mf > c. 9) We shall next show that P is globally asymptotically stable. Define hx, y) = 1. Clearly, h>0ifx>0 and y>0. xy Now, [ f 1 x, y) =x a 1 x ) cy ] [ ] fx, f 2 x, y) =y k my + x myx d, x, y) = x f 1h)+ y f 2h) = a cmf c) ky my + x) 2. Therefore, if mf > c, x, y) < 0. From Bendixson-Dulac criterion there exists no limit cycle and hence it is globally asymptotically stable in R 2 +.

6 390 Tapan Kumar Kar Prey 70 Predator 60 Populations Time Figure 2. Both the prey and predator populations converge to their equilibrium values. Remark. From the point of view of ecological managers, it may be desirable to have a unique positive equilibrium which is globally asymptotically stable, in order to plan harvesting and keep sustainable development of ecosystem. For example we consider the system dx dt =1.0x 1 x 100 dy dt = y [ x 5y + x ) 1.6xy 5y + x ] 0.06y. There is a positive equilibrium x,y ) = 73.87, 65.81). In this case, the behaviours of the prey and predator populations are shown in Figures 2 and 3. Case II. τ 0. We now study for which delay, the interior equilibrium becomes unstable. For τ 0, if λ = iw is a root of equation 7), then we have ω 2 + Fe iωτ + Aiω + C + ibωe iωτ =0. Separating real and imaginary parts we get C ω 2 + Bωsinωτ)+Fcosωτ) =0, Aω + Bωcosω t) F sinω t) = 0. 10) From 10), we obtain the fourth order equation for ω as ω 4 +A 2 B 2 2C)ω 2 + C 2 F 2 = 0 11)

7 Dynamics of a ratio-dependent prey-predator system P73.8,65.8) 60 Predator Prey Figure 3. Phase plane trajectories corresponding to different initial levels. From 11) it follows that if A 2 B 2 2C >0 and C 2 F 2 > 0, 12) then equation 11) does not have any real solutions To find the necessary and sufficient conditions for non-existence of delay induced instability we now use the following theorem [13]. Theorem 1. A set of necessary and sufficient conditions for x,y ) to be asymptotically stable for all τ 0 is the following: 1. The real parts of all the roots of F λ, 0)=0are negative 2. For all real m and τ 0,Fim, τ) 0, where i = 1. Theorem 2. If conditions 9) and 12) are satisfied, then x,y ) is locally asymptotically stable for all τ 0. Proof. Proof is obvious from conditions 9), 12) and the Theorem 1. From 11) we see that there is a unique positive solution ωo 2 if C 2 F 2 < 0. 13) Substituting ωo 2 into 10) and solving for τ, we get τ o n = 1 [ ω 2 arc cos o C)F ABωo 2 ω o F 2 + B 2 ωo 2 n =0, ±1, ±2,... Again if C 2 F 2 > 0, B 2 A 2 +2C>0and ] + 2nπ ω o 14) B 2 A 2 +2C) 2 > 4C 2 F 2 ) 15)

8 392 Tapan Kumar Kar hold, then there are two positive solutions ω± 2. Substituting ω2 ± into 10) and solving for τ, we obtain τ ± k = 1 arctan ω ±AF BC + Bω± 2 ) ) ω ± ABω± 2 +C ω2 ± )F + 2kπ,k =0, 1, 2,... ω ± 16) To prove theorems 3 and 4, differentiating Eq. 7) w.r.to τ we obtain [ 2λ + A + Be λτ τbλ + Fe λτ] dλ = λe λτ Bλ + F ). 17) Therefore ) 1 dλ 2λ + A = λλ 2 + Aλ + C) + λ by using e λτ 2 ) + Aλ + C =. Bλ + F Thus, { } { ) } 1 dreλ) dλ sign = sign Re λ=iω = sign B λbλ + F ) τ λ λ=iω [ 2ω 2 C)+A 2 ω 2 + C) 2 + A 2 ω 2 B 2 F 2 + B 2 ω 2 ] 18) 19) Theorem 3. If 9) and 13) hold, then the equilibrium x,y ) is asymptotically stable for τ < τ o and unstable for τ > τ o. Further, as τ increases through τ o, x,y ) bifurcates into small amplitude periodic solutions, where τ o = τ on as n =0. Proof. For τ =0, x,y ) is asymptotically stable if the condition 9) hold. Hence by Butler s lemma, x,y ) remains stable for τ < τ o. We have now to show that dreλ) τ=τo,ω=ω o > 0. This will signify that there exists at least one eigenvalue with positive real part for τ>τ o. Moreover, the condition of Hopf bifurcation [14] is then satisfied yielding the required periodic solution. From 19), it follows that { } { } dreλ) B2 A sign = sign 2 +2C) 2 4C 2 F 2 ) λ=iω o [ ωo 2 + C) 2 + A 2 ωo][f b 2 ωo] 2. Therefore dreλ) τ=τo,ω=ω o > 0. Therefore, the transversability condition holds and hence Hopf-bifurcation occurs at ω = ω o,τ = τ o. This completes the proof.

9 Dynamics of a ratio-dependent prey-predator system 393 Theorem 4. Let τ ± k be defined in 16). If9) and 15) hold, then there exist a positive integer m such that there are m switches from stability to instability and to stability. In other words, when τ [0,τ o + ), τ o,τ+ 1 ),..., τ m 1,τ+ m ), the equilibrium x,y ) is stable and when τ [τ o +,τ o ), τ 1 +,τ 1 ),..., τ m 1,τ+ m ), x,y ) is unstable. Therefore, there are bifurcation at x,y ) when τ = τ ± k,k = 0, 1, 2,...m. Proof. If conditions 9) and 15) hold, then to prove the theorem we need only to verify the transversability conditions see, [6, 14]). dreλ) τ=τ + k From 18), it follows that { } dreλ) sign = sign λ=iω + Therefore Again, sign { } dreλ) λ=iω > 0 and dreλ) τ=τ < 0. k { } B2 A 2 +2C) 2 4C 2 F 2 ) [ ω+ 2 + C)2 + A 2 ω+ 2 ][F 2 + B 2 ω+ 2 ]. dreλ) ω=ω+,τ=τ + k = sign { > 0. B 2 A 2 +2C) 2 4C 2 F 2 ) [ ω 2 + C)2 + F 2 ω 2 ][F 2 + B 2 ω 2 ] Therefore dreλ) ω=ω,τ=τ < 0. k Hence the transversability conditions are hold. This completes the theorem. 3. Discussion In this paper we have considered a ratio-dependent prey-predator model with stage-structure for predator and harvesting of adult predators. Our work indicates that, in the absence of delay, local stability of the positive equilibrium implies its global stability. On the other hand it has been found that if the positive equilibrium is stable at τ =0, then it would remain stable for all τ 0 under certain conditions. It has also been shown that the positive equilibrium, which is stable without delay, remains stable under certain conditions when the time delay is less than the threshold value, otherwise the stable equilibrium may become unstable. Finally, we have shown that there may appear switching of stabilities. As the maturation time delay τ increases oscillatory dynamics may appear and further increase of τ will return the oscillatory dynamics to the globally attractive steady state form. Ultimately, when the maturation time delay is too long, the positive steady state disappears and the predator population dies out. This shows that the sensitivity of the model dynamics on maturation time delay. }.

10 394 Tapan Kumar Kar The model that we have considered is one of several possible versions of the stage-structure model. It is an important problem in general to find tractable criteria to decide if a given model is a good model for a specific population growth situation. In this paper, we do not discuss this, but simply assume equation 5) as one of several natural models and then the dynamic behaviour of the system is studied. References 1. P. Abrams, The fallacies of ratio dependen t predation, Ecology ), F. Brauer and A. C. Soudack, Stability regions and transition phenomena for harvested predator-prey system, J. Math. Biol ), J. Chattopadhyay, G. Ghoshal and K. S. Chaudhuri, Nonselective harvesting of a preypredator community with infected prey, J. Appl. Math. and Computing old: KJCAM) 63) 1999), C. W. Clark, Mathematical Bioeconomics, The Optimal Management of Renewable Resources,2nd Eds., John Wiley & Sons., New York, J. M. Cushing, Integro-Differential Equations and Delay Models in Population Dynamics, Springer-verlag, Heidelber, J. M. Cushing and M. Saleem, A predator-prey model with age structure, J. Math. Biol ), R. Driver, Ordinary and Delay Differential Equations, New York, Springer, M. Fan and Y. Kuang, Dynamics of a nonautonomous predator prey system with the Beddington-DeAngelis functional response, J. Math. Anal. Appl ), M. Fan, Q. Wang and X. Zou, Dyanamics of a nonautonomous ratio-dependent predatorprey system, Proceeding of Royal Society of Edinburgh A ), M. Fan and K. Wang, Periodic solutions of a discrete time nonautonomous ratiodependent predator-prey system, Mathematical and Computers Modelling ), M. Fan and K. Wang, Periodicity in a delayed ratio-dependent predator prey systems, J. Math. Anal. Appl. 2621) 2001), R. Gambell, W. N. In, D. W. H Walton,Eds), Birds and Mammals-Antartic whales in Antarctica. Pergamon Press, Newyork, K. Gopalswamy, Stability and Oscillation in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht, J. K. Hale, Ordinary Differential Equations, Wiley, New York, T. K. Kar, Selective harvesting in a prey-predator fisher with time delay, Math. Comp. Model 383/4) 2003), T. K. Kar, Stability analysis of a prey-predator model with delay and harvesting, J. Biol. Syst. 121) 2004), T. K. Kar, Influence of environmental noises on the Gompertz model of two species fishery, Ecological Modelling ) 2004), T. K. Kar, Optimal harvesting and stability for a prey-predator system with stage structure. Advances in Modelling, Series D, AMSE Periodicals 83) 2003), Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol ), Y. Kuang, Rich dynamics of Gause-type ratio-dependent predator-prey system, Field Institute Communications ), N. MacDonald, Time Lags in Biological Models, Springer-Verlag, Heidelberg, 1978.

11 Dynamics of a ratio-dependent prey-predator system A. Mukhopadhyay, J. Chattopadhyay and P. K. Tapaswi, Selective harvesting in a two species fisher model, Ecol. Model ), G. P. Samanta, D. Manna and A. Maiti, Bioeconomic modelling of a three-species fishery with switching effect, J. Appl. Math. & Computing ), X. Song and L. Chen, Optimal harvesting and stability for a two species competitive system with stage structure, Math. Biosci ), C. Zhang, M. Liu and B. Zheng, Hoft bifurcation in numerical approximation for delay differential equations, J. Appl. Math. & Computing ), 319. T. K. Kar received his M. Sc. and M. Phil. from Calcutta University and Ph. D. from Jadavpur University. Since 1996 he has been at the Bengal Engineering and Science University [formerly Bengal Engineering college a Deemed University)] as a Lecturer of Mathematics. His research interest is Mathematical Ecology, particularly Bioeconomic Harvesting Problem and Epidemics. Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah , India t k kar@yahoo.com