Hunting Co-operation in the Middle Predator in Three Species Food Chain Model
HHunting Co-operation in the Middle Predator in ThreeSpecies Food Chain Model
Nishchal Sapkota , , Rimsha Bhatta , Phillip Dabney Advisor: Dr. Zhifu Xie School of Mathematics and Natural Sciences School of Computing Sciences and Computer Engineering School of Polymer Science and EngineeringThe University of Southern MississippiHattiesburg, Mississippi 39406The United States of Americaemail: [email protected], [email protected],[email protected], [email protected]
Abstract
We proposed a three-species food chain model with hunting co-operation among the middle preda-tor. In this model, third species prey on the middle species and the middle prey on the first species.The hunting cooperation among the middle predator affects interestingly on the numbers of both thepredators and the prey. We examined the linear stability of the model theoretically and numerically .We conducted the two-parameter numerical analysis to check the long-term behavior and the change inthe number of the species with respect to hunting co-operation. Our findings supported the postulatesfrom the two species food chain model with hunting co-operation.Keywords: Predator-prey model, hunting co-operation, three species food chain model.
One of the ways to describe the ecological system is through a food chain/web model between predatorand prey. In mathematics, the classical predator-prey model has been analyzed by constructing a math-ematical model to help us solve the fundamental biological problems. The interaction between the threespecies on a linear food chain model is well studied in a three-dimension [1]. One of the natural and realisticcases of a simple food chain model is the one where the top predator is a generalist and the middle predatoris a specialist [1]. A generalist predator is defined as the predator who can survive even in the absence ofits special food, whereas a specialist predator is the one which can survive only in the presence of specialor favorite food. The specialist predator dies in the absence of specific food. A special parameter, α , isintroduced in the model. α is the hunting cooperation under which, the variation in dynamical behavioror the chaotic behavior is studied. Originally, this had been studied by Upadhyay et. al [3] depending onthe various functional response.Animals hunt together in groups under the division of labor and specified roles [2]. This kind of behav-ior involves two or more animal-eating species, successfully capturing a common prey, which helps all theinvolved individual to minimize the cost of time and effort than when alone. Here, the three species foodchain model is combined with the hunting co-operation to see the changes in the long term behavior of thepopulation. Thereby, the variation in dynamics brought by the initial value condition of hunting coopera-tion under several constant parameters is studied and their long -term behavior is noted. The mainstreamof the paper is to focus on the effect of hunting cooperation in three species models. Numerical methods1 a r X i v : . [ q - b i o . P E ] J un o solve the system of differential equations, six-stage fifth-order Runge-Kutta method, is implemented inMATLAB.The relationship between hunting cooperation and growth rate is modeled by keeping all the otherparameters constant. Our main goal is to analyze how the different intensities of hunting co-operationbring changes in the population density, survival of species and the stability of the ecosystem dynamics. The illustrated three species food chain model is the combination of a three species model and Huntingcooperation. It is modified from the classical Upadhyay-lyengar-Rai model by adding hunting cooperationin the specialist predator [2]. du dt = u ( a − b u − w (1+ αu ) u (1+ αu ) u + D ) , du dt = u ( − a + w (1+ αu ) u (1+ αu ) u + D − w u u + D ) du dt = u ( a − w u + D ) (1)Here, u , u and u denote the three species, prey, middle predator and top predator respectively. α denotes the hunting cooperation. b is the intra-species competition in the prey. a denotes the growthrate of u , a denotes the rate at which u dies out in absence of u and u , and a is the growth rateof u through sexual reproduction. w is the maximum value which per capita reduction rate can attain.In other words, it is the maximum predation that can occur among the species. D and D signifies themaximum limit to which environment provides protection to the prey u . D is the value of u at whichits per capital removal rate becomes w and D depicts the residual loss in u in absence of its favoritefood u [3].Analysis of this system is an interesting problem as hunting co-operation has barely been analyzed inthree species model. This will also help us draw parallels with the similar work done in two species modelby Teixeira et.al [2]. Furthermore, we could extend the dynamics of three species model explained in Haileand Xie’s work [1]. Rewriting the system of equations (1) in product form and setting them equal to 0. u G ( u , u , u ) := u ( a − b u − w u u + D ) = 0 u G ( u , u , u ) := u ( − a − b u + w u u + D − w u u + D ) = 0 u G ( u , u , u ) := u ( a − w u + D ) = 0 (2)Upon solving (2) we get the following equilibrium points: u [1] = (0 , , u [2] = ( a b , , u [3] = ( u +1 , u , u +3 ), u [4] = ( u − , u , u − ),where, u ± = ( a H − b D ) ± √ ( b D − a H ) − b H ( w u H − a D b H , u = w a − D , u ± = ( u + D ) w (cid:104) w u ± H ( u ± H + D ) − a (cid:105) , 2or H = (1 + αu ).However, for some choices of parameters, u − and u − yield either negative or non real values, hence, weavoid u [4] in our analysis. Similarly, u [1] is trivial, and u [2] doesn’t offer much either. Hence we will focusmore on u [3] . u [3] To study the linear stability of the model, the Jacobin matrix ( J ) is calculated by taking the partialdifferentiation of system of equations (2), with respect to u , u , u respectively. J = J J J J J J J J J Where, J = G + u [ − b + w u (1+ αu ) [(1+ α + u ) u + D ] ], J = − u [ w ( u + D )+ αw [2( u + D ) u + αu u ][ u + D + αu u ] ], J = 0 J = u [ D w (1+ αu )[(1+ αu ) u + D ] ], J = G + u [ αu w D [(1+ αu ) u + D ] + w u [ u + D ] ], J = − u [ w u + D ], J = 0, J = ( u ) [ w ( u + D ) ], J = 2 u G Also, J is almost equal to zero for the provided parameters.Thus, J = J J J J J J Theorem 2.1
Assuming the inequalities J − J < and J J > J J hold when the positive param-eters in model (1) are in a set Γ , if u [3] is a positive coexistence equilibrium solution and the parametersare in the set Γ , then it is linearly stable.Proof: From this matrix, the reduced form of the characteristics polynomial is given by the followingexpression. [1] p ( λ ) = λ + A λ + A λ + A (3)where, A = − J − J , A = J J − J J − J J , A = J J J . Using Routh-Hurwitz criterion for third order polynomials, equation (3) will have a stable solution if, A > , A > A A > A . The positive terms are J , J , J and the negative terms are J , J .Therefore, J has to be negative to satisfy A > A is zero for some choices of parameter, our characteristics polynomial is further reducedto a second degree expression of the form: p ( λ ) = λ + A λ + A (4)3ow, using the Routh-Hurwitz criterion for second order polynomials, equation (4) will have a stablesolution if A > A >
0. We have already reached to the conjecture that A >
0. And, for A > J J > J J + J J has to be satisfied.Here, the resultant quantity on the right hand side of the inequality is negative as, J and J arenegative. Because J is positive, J is either a positive number or a negative number such that its productwith J is greater than the right hand side. To observe how hunting co-operation would bring changes to the long term behavior of the system, weuse ode45 function in MATLAB to solve the system (1).Values of the parameters are chosen referring tothe standard values as used in reference [1]: a = 1 , a = 0 . , b = 0 . , w = 1 , w = 0 . , w = 1 , D =10 , D = 10 , D = 10 , D = 20.Figure 1: Variation of u i with α ( i = 1 , ,
3) at a = 1 . u +1 , u and u +3 in section 2.1 with the value of a being 1.6.Our concerns for α is within 0 and 1, where any value of α close to 1 indicates the maximum huntingco-operation. The higher value of α could affect the discriminant of u ± , and hence the resultant valuesfor u i could either be negative or non real values. Here, u seems to be unaffected with any degrees ofco-operation involved while u and u seem to be converging to a fixed number as hunting co-operationreaches it maximum value. 4or the results below, the initial values for ( u , u , u ) was kept as (15 , , α and a accordingly, to observe the long term behavior of the system. a < . Figure 2: ( a = 0 .
3) : u stable, u and u extinctFigure 3: ( a = 1): u and u stable, u extinctFigure 4: ( a = 1 . α did not change the dynamics of the system for small value of a . However, forhigher value of a , α changed the stability of the system. As the value of α increased, the stable attributesin the graph changed. Stable u and u was made unstable in figure 2, while the co-existence state of u , u , and u in figure 3 were also made unstable. In the first two cases, u was independent of the change of α . However, in the third case, u changed from stable co-existence and ultimately seemed to extinct.5 .2 Cases for a ≥ . Figure 5: ( a = 1 . a = 1 . a = 2): ChaosFigure 8: ( a = 2): Deformation of ChaosLimit oscillation is observed for a = 1 .
5, and chaos is observed for a = 2 in the absence of huntingco-operation ( α = 0). As we increased the value of α , both of these were deformed. The amplitude oflimit cycle increased as we increased the value of α . The deformation was very sensitive even for the smallchanges in α . 6 CONCLUSION AND FUTURE DIRECTION
From the numerical results above, we can conclude that hunting co-operation significantly changes thebehavior of the prey and the middle predator for higher values of growth rate of the prey. However, theeffect of hunting co-operation was minimal when the value of growth rate of prey was smaller. Huntingco-operation leads to the extinction of top predator even for the co-existence equilibrium. It changes thestability of the coexistence equilibrium. Limit cycle oscillations emerges and the amplitude of oscillationsincreases with hunting co-operation.The results of our work draws parallels to the postulates of hunting co-operation in two species model[2]. Further, we aim to study Allee Effects with an anticipation that Alee threshold could occur on theboundary between basin of attraction of co-existence state and basin of attraction of predator-extinctionstate [2]. Identifying the stable and unstable region in graph of top predator and the prey, we could separatethe basin of attraction. This could potentially verify that Allee Threshold varies with prey population andfor higher prey population, the hunting co-operation leads to smaller Allee Threshold thereby reducing therisk of predator extinction.
We are grateful to MAA National Research Experiences for Undergraduates Program in the Math-ematical Sciences (NREUP) and Wright W. and Annie Rea Cross Endowed Chair in Mathematics andUndergraduate Research for funding us to participate in this program. We are also thankful to our men-tor, Dr. Zhifu Xie for guiding us through this process.
References [1] Dawit Haile, ZhifuXie.
Long-time behavior and Turing instability induced by cross-diffusion in a threespecies food chain model with a Holling type-II functional response . Mathematical Biosciences. 267(2015)134-148.[2] Mickael Teixeira Alvesa, Frank M. Hilker.
Hunting cooperation and Allee effects in predators.
Journalof Theoretical Biology. 419 (2017) 13?22.[3] R.D. Parshad, R.K. Upadhyay.