On Prey-Predator with Group Defense

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1 ISSN (print) (online) International Journal of Nonlinear Science Vol.15(013) No.4pp On Prey-Predator with Group Defense Ali A Hashem 1 I. Siddique 1 Department of Mathematics Faculty of Science Sana a University P. O. Box 1157 Yeman. 1 Department of Mathematics Faculty of Science Jazan University Jazan P. O. Box 097 Kingdom of Saudi Arabia. Department of Mathematics COMSATS Institute of Information Technology Lahore Pakistan. (Received 5 September 01 accepted 13 March 013) Abstract: A mathematical model with one prey species living in two different habitats and a predator a prey exhibits group defense is studied. The prey species is able to migrate between two different habitats e to change in seasonal conditions. The stability analysis is carried out for a critical point of the system all species co-exist. Using rate of conversion of the prey to predator as bifurcation parameter conditions for a Hopf bifurcation to occur are derived. Keywords: prey-predator; group defense; stability analysis 1 Introction In predator-prey environment the predator prefers to feed-itself in a habitat or some ration and changes its preference to another habitat. This preferential phenomenon of change of habitats from one to other is called switching. The mathematical models that have generally been proposed with switching are those involving interaction of one predator with two prey species (Tansky [1] Prajneshu and Holgate [] Khan et al. [3 5] Song et al. [6]) and two predator and two species Qamar et al. [7]. Large prey species like wildebeest Zebra Thomsan s gazelle feed upon abundant evenly disperse easily find food items. Food items of low quality (nature leaves stems) are much more in abundant than those of high quality within selected habitats and so they form huge and cohesive groups. Seasonal condition of heavy rain storms dry season and other harsh environment force wildlife to migrate to another habitats for better condition food and surface water. Usually wild life animals distillate near permanent water ring dry periods and disperse into neighboring dry habitat ring wet periods. With the advent of dry weather the wildebeest return to their dry seasons range because grass growth on the wet seasons range stops after a few days without rain and there remains almost no standing crop as for reservoir. Free standing water is also largely absent from this area. Wildebeests are thus forced to return to their dry season range which maintain green leaf for a long period and retains substantial reservoir of grass swards e to light grazing pressure in the wet season. Free standing water is also available there in pools among major river systems. In this way herbivores maximize the growth potential of the vegetation through the rotational grazing the two concentration areas are sufficiently far apart the movements are called seasonal migration. Large herd farmers are more likely dependent on self defense and group defense to avoid being killed by predators. Group defense is a term used to describe a phenomenon by predation is decreased or even prevented altogether by the ability of the prey population to better defend themselves when their number is large. The predators do not seek out areas with very large prey density. Pairs of musk-oxen can be successfully attacked by wolves but groups are rarely attacked. There are many examples of group defense. Herds remain well coordinated even under attack and indivials may benefit from the alertness and communication. Indivials tend to conform with their neighbor activities and many hundreds even thousands of wildebeest can coordinate rapidly in response to an alarm. Large groups also benefit from increased probability of detection of predators. The hunting success of lions decline if the group size of prey is large. Cheetah prefers to hunt single animals. Corresponding author. address: imransmsrazi@gmail.com Copyright c World Academic Press World Academic Union IJNS /77

2 9 International Journal of Nonlinear Science Vol.15(013) No.4 pp (Tansky [1]) investigated a mathematical model of two prey and one predator system which has the switching property of predation of the following from dx dz = = { } az r (y/x) n x { } bz r 1 + (y/x) n y = r 3 + a 1 xz 1 + (y/x) n + a yz n = (y/x) n x y and z denote the abundance of two kinds of the prey species and predator species respectively r 1 and r are the specific growth rates of the prey species in the absence of predation and r 3 is the pre capita death rate of the predator. a b The functions 1+(y/x) and n 1+(y/x) have a characteristic property of switching mechanism. The predatory rate that an n indivial of the prey species is attacked by a predator decreases when the population of that species becomes exceptional compared with the population of another prey species. This property is much amplified for large value of n. He studied the model with simplest form of the function for n = 1. The model The prey-predator model the pre species exhibits group defense is of the form dx 1 =α 1 x 1 ε 1 x 1 + ε P 1 x β 1x y ( ) n x x dx =α x ε x + ε 1 P 1 x 1 β x 1 y ( ) n x 1 + x 1 [ µ = + δ 1β 1 x n+1 x n δβ x n+1 ] 1 xn x n 1 + y xn (1) with n = 3 the above equations become dx 1 dx = =α 1 x 1 ε 1 x 1 + ε P 1 x β 1x 4 y x x3 =α x ε x + ε 1 P 1 x 1 β x 4 1y [ µ + δ 1β 1 x 4 + δ β x 4 1 x x3 x x3 x x3 ] y () with x 1 (0) > 0 x (0) > 0 y(0) > 0. x i : represents the population of the prey in two different habitats. y: represents the population of predator species. β i : measure the feeding rates of the predator on the prey species in habitat 1 and habitat δ i : conversion rate of prey to predator ε i : inverse barrier strength in going out of the first habitat P ij : the probability of successful transition from i th to j th habitat (i j) α i : pre capita birth rate of prey species in two different habitats µ: the death rate of the predator. For n = 1 the case has been studied by Khan et. al [5] the relative abundance of the prey species has a simple multiplicative effect. For n = the case has been studied by Khan et. al [8] the effect of relative density is stronger than the simple multiplicative. Here we have studied the same model with n = 3. IJNS for contribution: editor@nonlinearscience.org.uk

3 Ali A Hashem I. Siddique: On Prey-Predator with Group Defense 93 3 Stability of equilibria The non-zero equilibrium point of the system () is given by: or x 1 = µx(x3 + 1) δ 1 β 1 + δ β x 4 x = µ(x3 + 1) δ 1 β 1 + δ β x 4 y = ((α 1 ε 1 )x + ε P 1 )(x 3 + 1) β 1. y = ((α ε ) + ε 1 P 1 x)(x 3 + 1) β x 4 x = x 1 /x is a real positive root of the following fifth order equation β (α 1 ε 1 )x 5 + β ε P 1 x 4 β 1 ε 1 P 1 x β 1 (α ε ) = 0. (4) For equilibrium values (x 1 x y) be a positive real root of the Eq. (4) must be bounded therefore ε α ε 1 P 1 < x < ε P 1 ε 1 α 1. (5) Let E = (x 1 x y) denote the non-zero equilibrium point x 1 x y > 0. We investigate the stability of E and the bifurcation structure particularly Hopf bifurcation for the system (). Using δ i (conversion rates of the prey u to the predator) as the bifurcation parameter. We first obtain the characteristic equation for the linearization of the system () near the equilibrium. We consider a small perturbation about the equilibrium value i.e x 1 = x 1 + u x = x + v and y = y + w. Substituting these values into the system () and neglecting the terms of second order in small quantities we obtain the stability matrix equation which leads to the eigenvalue equation a 1 = (A + D) a = F β x 4 1 x x 3 + Eβ 1x 4 x x 3 a 3 = β 1x 4 x 3 1 +x3 A µ B C D µ βx4 1 = 0 (6) x 3 1 +x3 E F µ µ 3 + a 1 µ + a µ + a 3 = 0 (7) ( DF β1 x 5 x 1 (x x 3 ) + DEβ 1x 4 x x 3 + AF β x 4 1 x x 3 + AEβ x 5 1 x (x x 3 ) A =(α 1 ε 1 ) + 3yβ 1x 1x 4 (x x 3 ) ) (3) B = Ax 1 = ε P 1 + 3yβ 1x 6 x (x x 3 ) 4yβ 1x 3 x x 3 (8) C = Dx = ε 1 P 1 + 3yβ x 6 1 x 1 (x x 3 ) 4yβ x 3 1 x x 3 D =(α ε ) + 3yβ x 4 1x (x x 3 ) E = 4yδ β x 3 1 x x 3 F = 4yδ 1β 1 x 3 x x 3 3yδ β x 6 1 (x x 3 ) 3yδ 1β 1 x 1x 4 (x x 3 ) 3yδ 1β 1 x 6 (x x 3 ) 3yδ β x 4 1x (x x 3 ). IJNS homepage:

4 94 International Journal of Nonlinear Science Vol.15(013) No.4 pp The Routh-Hurwitz stability criteria of eigenvalue equation (7) is a 1 > 0 a 3 > 0 and a 1 a > a 3. Hence the equilibrium E will be locally stable to small perturbation if it satisfies the following conditions: and α 1 + α + 3y x 1 x (x x 3 ) (β 1x 4 + β x 4 1) <ε 1 + ε (Dβ 1 + Aβ x 5 )(E + F x ) <0 (9) (β 1 + β x 5 )(AEx + DF ) < 0. (10) Stability of the equilibrium point depends upon the conditions (5) and (9) together with various parameters. 4 Hopf bifurcation analysis We stu the Hopf bifurcation for the system of () using δ 1 (rate of conversion of the prey in habitat 1 to the predator) as the bifurcation parameter. The eigenvalue equation (7) has purely imaginary roots if and only if a 1 a = a 3 for some value of δ 1 (say δ 1 = δ 1). For A < 0 D < 0 E > 0 and F > 0; a 1 a a 3 are positive. There exists δ 1 such that a 1 a = a 3. Therefore there is only one value of δ 1 at which we have a bifurcation. For some ε > 0 for which δ 1 ε > 0 there is a neighborhood of δ 1 say (δ 1 ε δ 1 + ε) in which the eigenvalue equation (7) cannot have real positive roots. For δ 1 = δ 1 we have (µ + a )(µ + a 1 ) = 0 (11) which has the three roots The roots are in general of the form µ 1 = i a µ = i a and µ 3 = a 1. µ 1 (δ 1 ) =u(δ 1 ) + iv(δ 1 ) µ (δ 1 ) =u(δ 1 ) iv(δ 1 ) µ 3 (δ 1 ) = a 1 (δ 1 ). (1) To apply the Hopf bifurcation theorem as stated in (Marsden and McCrachen [9]) we need to verify the transversality condition δ1 =δ dδ 0. (13) 1 1 Substituting µ k (δ 1 ) = u(δ 1 ) + iv(δ 1 ) into the equation (7) and differentiating the resulting equations with respect to δ 1 and setting u = 0 and v(δ 1 ) = v 1 we get ( 3v 1 + a ) + dv ( a 1 v 1 ) =á 1 v 1 á 3 ( a 1 v 1 ) + dv ( 3v 1 + a ) = á 1 v 1 á 1 = da 1 = 0 (because A and D are independent of δ 1 ). á = da and á 3 = da 3 x is a real positive root of the equation (4) which is independent of δ 1. Solving for and dv we have δ1 =δ dδ = a (a 1 á á 3 ) 1 1 4a + 4a 1 a. (15) To establish the Hopf bifurcation at δ 1 = δ 1 we need to show that At δ 1 = δ 1 a 1 a = a 3 ; gives (14) δ1 =δ 1 0 i.e a 1á á 3 0. (16) (β 1 β x 5 1)(AEx 1 DF ) = 0 IJNS for contribution: editor@nonlinearscience.org.uk

5 Ali A Hashem I. Siddique: On Prey-Predator with Group Defense 95 i.e x = DF AE. (17) Substituting the values of a 1 á and á 3 in the equation (16) and using the equation (17) we get a 1 á á 3 = x 1 + x 3 (β 1 β x 5 ) provided x 5 β 1 /β and µ 3 (δ 1 ) = a 1 (δ 1 ) 0. We summarize the above results in the following theorem: ( D df A de x D df A de = AE d(f/e) < 0 x F i.e δ1 =δ 1 > 0 ) Theorem 1 Suppose E = (x 1 x y) exists A < 0 D < 0 E > 0 F > 0 and δ 1 be a positive root of the equation a 1 a = a 3 then a Hopf bifurcation occurs and δ 1 passes through δ 1 provided β 1 /β x 5. Similar analysis can be carried out by varying δ (rate of conversion of the prey in second habitat to the predator) and we shall get the similar results. References [1] M. Tansky. Switching effects in prey-predator system. J. Theor. Bio 70(1978): [] Pranjneshu and P. Holgate. A prey model with switching effect. J. Theor. Bio 15(1987): [3] Q. J. A. Khan and B. S. Bhutt. Hopf bifurcation analysis of a predator-prey system involving switching. J. Phys. Soc. Jpn 65(1996): [4] Q. J. A. Khan B. S. Bhutt and R. P. Jaju. Stability of a switching model with two habitats and predator. J. Phys. Soc. Jpn 63(1994): [5] Q. J. A. Khan B. S. Bhutt and R. P. Jaju. Switching model with two habitats and a predator involving group defense. J. Non. Math. Phys 5(1998):1-9. [6] X. Song and L. Chen. Optimal harvesting and stability for a two-species competitive system with stage structure. Math. Biosci 170(001): [7] Q. J. A. Khan Edamana V. Krishnan and E. Balakrishnan. Two-predator and two-secies group defence model with switching effect. Int. J. Comput. Biosci 17(010): [8] Q. J. A. Khan B. S. Bhutt and R. P. Jaju. Switching effects of predator on large size prey species exhibiting group defense. Differ. Eqs. Cont. Proc. Elec. J 3(1999): [9] J. E. Marsden and M. McCracken. The Hopf Bifurcation and its applications. Springer-Verlage New York IJNS homepage: