Existence, Uniqueness Solution of a Modified. Predator-Prey Model
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1 Nonlinear Analysis and Differential Equations, Vol. 4, 6, no. 4, HIKARI Ltd, Existence, Uniqueness Solution of a Modified Predator-Prey Model M. A. Al Qudah Mathematical Science Department Princess Nourah Bint Abdulrahman University P.O. Box 8448, Riyadh 67, Saudi Arabia Copyright 6 M. A. Al Qudah. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract A modified predator-prey model which describes the interaction between the Predator-Prey species and at the species itself is considered. The existence and uniqueness of solutions of a modified predator-prey model in L p,q space is proved under certain conditions of p and q. Some properties of the dynamically system of such model are described such as its stability. Graphs of solutions displayed. Applications of the results in mathematical biology are discussed. Mathematics Subject Classification: 9B99 Keywords: Predator-Prey model, existence, uniqueness, Stability. Introduction The dynamics of population has been described using mathematical models which have been very successfully for studying animals and human populations. Lotka [] initiated the predator-prey models and competing species relations. Levin and Segel [] studied some biological hypotheses concerning the origin of Planktonic Patchiness model. Fife [] considered reaction and diffusion systems which are distributed in 3-dimentional space or on a surface rather than on the line. Calderon [9] studied the diffusion and non-linear population theory. Abualrub [] discussed diffusion problems in mathematical biology. In addition, Abualrub [] studied diffusion in - dimensional spaces for which diffusion is more realistic and applicable in life and he proved the existence and uniqueness of long range diffusion reaction model on population dynamics. Abualrub [4] proved
2 67 M. A. Al Qudah the existence and uniqueness of solutions of a diffusive predator-prey model. Al- Qudah and Abualrub [5] studied the existence and uniqueness of long range diffusion involving flux for insect dispersal model, and they studied in [7] solitary and traveling wave solutions with stability analysis also for insect dispersal model. In this paper, we make another modification of the model of Abualrub [3] which is a modification to planktonic patchiness model, - a kind of predator-prey model-, for more details see []. Existence and uniqueness of solutions for the modified model will be considered. A traveling wave solutions and stability have been considered. This paper is organized as follows. In Section, a modified Predator- Prey model is considered. The existence and uniqueness of its solutions is given in Section 3. The stability analysis and the conclusion are given in Section 4.. The predator- prey model The model we are going to consider here is another modification to the planktonic patchiness model, a kind of predator-prey models, which were originally considered by Levin and Segel in 976 []. We use the model in [4, 3, 6] with the assumption that the species specific diffusion coefficients be constant to come up with the predator- prey model: u t u = a u + a u a 3 uv, () v t v = a 4 uv a 5 v. () Where u = u(x, y) is the prey density population, v = v(x, y)is the predator density population, x = (x, x ), and = x + x represents the diffusion (dispersal), we shall assume that a,,a 5 are constants and a may assumed to be compact supported and bounded function of x; that is ( a = a (x); a (x) = if x > N; where N; is a constant ) not a constant this is assumed, because the birth (or death) rate may depend on the environment, which is assumed to be bounded. Another reason for assuming that a,,a 5 are constants, is due to the fact that the birth(or death) rate depends on the interaction between the male and the female (sexual interaction as in terms a u and a 5 v ) or on the binary interaction between the males and females of the prey and the predator respectively ( as in the term uv). In this paper we assume that a is constant and we modify the model by assuming another kind of interaction between the prey and the predator, that is two terms are added, namely to get the following model : u t D u = a u a u a 3 uv a 4 uv, (3) v t D v = a 5 uv a 6 v a 7 v + a 8 uv. (4) The added terms a 4 uv, and a 8 uv mean that there exists an interaction between the species itself such as (in the mating period so we can consider the term v is the existence of a male and a female together, this will lead two predators
3 Existence, uniqueness solution of a modified predator-prey model 67 to meet on a prey. As the predator has a strong rapacity and can take more than it needs so it is better to consider the term uv ), and then interact the two species together in the environmental. The constants a,a,,a 8 are positive and D, D are the diffusion coefficients which are small positive constants. Finally assume that the initial data are in the same L p space for some p >, The initial values for Eqs. (3) and (4) will be given by u(x, ) = g(x), v(x, ) = h(x) respectively, where both g(x) and h(x) L p ( ). In addition we will consider small values of time t, since we are looking for local solution in the usual diffusion, but for large values of time one should talk about long range diffusion as in [, 5]. 3. Usual diffusion with p = q in the L p,q norms To easily solve Eqs. (3) and (4) we shall make the terms a u and a 6 v disappear from Eqs. (3) and (4); to do this let u(x, t) = e αt w(x, t), and v(x, t) = e βt z(x, t) where α = a, β = a 6. Therefore Eqs. (3) and (4) together with the initial data become as follows: w t D w = a e αt w a 3 e βt wz a 4 e βt wz, (5) w(x, ) = g(x), x, (6) z t D z = a 5 e αt wz a 7 e βt z + a 8 e (α+β)t wz, (7) z(x, ) = h(x), x, (8) since we have the heat operator in the left hand side of Eqs.(5) and (7), t therefore w and z can be obtained by solving the following integral equations: t w = K (x y, t τ)[ a e ατ w a 3 e βτ wz a 4 e βτ wz ] dydτ t + K (x y, t) g(y)dy, (9) z = K (x y, t τ)[a 5 e ατ wz a 7 e βτ z + a 8 e (α+β)τ wz ] dydτ + K (x y, t) h(y)dy, () where K, K are the fundamental solutions of the heat equation; thus: K (x, t) = D t and K πt (x, t) = D t, x = (x πt + x ), t >. Let D = max{d, D } where D is a small positive constant. And K, K can be estimated as in []; then
4 67 M. A. Al Qudah K, (x, t) D ( x +t ). () Using the symbol to represent the convolution in space and time while the symbol is to represent the convolution in space only; we can rewrite Eqs. (9) and () in a simpler way as follows: w = K [ a e ατ w a 3 e βτ wz a 4 e βτ wz ] + K g, () z = K [a 5 e ατ wz a 7 e βτ z + a 8 e (α+β)τ wz ] + K h. (3) where w and z are weak solutions of Eqs. (), (), (3) and (4) respectively, which implies that the integrals in Eqs. (9) and () exist in the Lebesgue sense. Lemma If w(x, t), z(x, t)εl ( 6, 6 ) ( [,]); and g(x), h(x)εl ( 4 ) ( ), then for ε >, T(w) 6, 6 T(z) 6, 6 C 6 + C 6, 6, 6 A 6 + A 6, 6, 6 + C 6 z 8, 8 z 6, 6 z 8, 8, 6 z 6, 6 + C 4 A 6 + A 4 g 4, and z 8, 8 h 4. Proof : Consider T(w) and T(w) to be the image of w and z respectively, such that: T(w) = K [ a e ατ w a 3 e βτ wz a 4 e βτ wz ] + K g, (4) T(z) = K [a 5 e ατ wz a 7 e βτ z + a 8 e (α+β)τ wz ] + K h. (5) Assume that t takes small values in order to show the existence and uniqueness of local solutions to Eqs. (4) and (5). Take the first, second, third, and fourth terms on the right hand side of Eq. (5) we shall use exponents r, s, p, q respectively, when considering the L p norm and we have the same argument for Eq. (4). It is obvious from Eq. () that: D K (x, t) ( x + t ; (6) ) using Eq. (6) and the same estimation of K as in [7] which is given by: K (x, t) Dε x θ t (θ+ε) (7)
5 Existence, uniqueness solution of a modified predator-prey model 673 where t. T is very small, < θ < and the very small positive constant ε is chosen such that < θ + ε <. From Eqs. (4) and (7), we obtain: T(z(x, t)) Da 5 e αt ε +Da 8 e (α+β)t ε Da 7 e αt ε w(y, τ) z(y, τ) dydτ x y θ t τ (θ+ε) w(y,τ) z(y,τ) dydτ x y θ t τ (θ+ε) z(y, τ) dydτ x y θ t τ (θ+ε) + K h. (8) Now, assume that hεl p ( ), z εl q ( ), w z εl q ( ), wz εl q ( ). Let r > be chosen such that: r = p θ ; < p < 4 θ, (9) by taking the L r ( ) norm of the both sides and use the Benedek-Panzone Potential Theorem, see [8], the first, the second and the third terms of the right hand side of Eq. (8) become: T(z(., t)) r Da 5 d p ε w(., τ) p z(., τ) p dτ t τ (θ+ε) Da 7 d p ε z(., τ) p dτ + Da 8 d p ε w(., τ) p z(., τ) p dτ t τ (θ+ε) t τ (θ+ε) + K h(., t) r. () Let w(., τ) p z(., τ) p εl q (R + ), z(., τ) p εl q (R + ), and w(., τ) p z(., τ) p εl q (R + ), and lets > such that for the third term we have: s = 3 (θ + ε) 6 ; 3 < q < q (θ + ε), () Again, by applying the Benedek-Panzone Potential Theorem, the first, the second and the third terms of the right hand side of Eq. (), and taking the L s (R + )norm of the both sides, we obtain: T(z) r,s Da 5 d p d q w r,s z p,q Da 7 d p d q z p,q + Da 8 d p d q w p,q z p,q + K h r. () Now, take p = r, then from Eq.(9) we get p = and q = s, then from Eq.() θ 4 we have q =, < θ <, ε >. In addition we shall require that p = q. (θ+ε)
6 674 M. A. Al Qudah Therefor θ = ; which implies that: = q = r = s = 6, and selecting 3 A 6 = Da 5 d p d q, A 6 = Da 7 d p d q, and A 6 = Da 8 d p d q. And since K h 6 follows directly from imbedding Lemma (namely Lemma ) for the A 4 h 4 initial data, this turns will conclude the main Theorem. Lemma Let F(x, t) = K h. Assume that h L r ( ), < r <. If for p, q > with + =, then F(x, t) p q r Lp,q ( R + ) and F p,q c(r) h r, where c(r) is a constant depending on r and the dimension. Proof see [4]. Theorem 3 Suppose that the initial data g, h L(R 4 ), < r <, < θ <, if ε > is very small such that g 4 h 4 < ε, then a unique solution u, v such that u 6, 6, v 6, 6 < ; where < ε <. The proof of this theorem can be done using the same argument as the proof of Lemma. 4. Stability Analysis In this section we are discuss the stability of the modified model given by Eqs. (3), (4) in one dimension. First we dimensionless the system by setting U = a u, V = a 3 v, T = a a a t, X = ( a ) x, D = D, k D D = a 5, k a = a 6, a a k 4 = a 3, a a 7 = a 3 k, a 8 = a 3a 5, then we have: a U T = DU XX + U U UV UV, V T = k V[U k V + UV] + V XX. (3) It is clear that k, k are positive constants since a, a,, a 8 are positive using the fact that only the predator are capable of moving toward the prey we have D = and letting U(X, T) = u (ξ) and V(X, T) = v (ξ): ξ = X ct where c is the wave speed which must be determined, thus system (3) become: c du dξ = u [ u v v ], c dv dξ = k v [u k v u v ] + d v dξ, (4) Using the method of reduction of order by setting m= dv then we have a system of dξ nonlinear of first order ODE's:
7 Existence, uniqueness solution of a modified predator-prey model 675 du dξ = u c [ u v v ], dv dξ = m, dm dξ = cm k v [u k v u v ]. (5) It is clear that system (5) is an almost linear system and has equilibrium points of the form (u, v, m) by setting du = dv = dm = the equilibrium points dξ dξ dξ are:(,,), (,,) and (, k, ). The Jacobian matrix corresponding system (5) is: [ u v v ] [u +u v ] c c J = [ ]. k v + k v k u + k k + k v k u v c It is clear that the points (,,) and (,,) are unstable saddle points since the corresponding eigenvalues of the characteristic equation for each equilibria are of opposite signs, but the point (, k, ) is asymptotically stable because its characteristic equation: ( k k c roots: λ = k k c λ) (λ + cλ + k k ) = has the following < if k ε [, + 5 ] since k and c are positive constants. λ,3 = c± c 4k k < if c > k k then the point is node, and if c < k k then the point is spiral in each case the point is asymptotically stable. Conclusion A- The wave speed c depends on the a, a, a 5 and a 6 actually on a 4 and a 8 (since a 5 = a 8 a 4 a 3 ) which are related to the coefficients of uv and uv. Thus the coefficients of uv in the prey equation is less than the coefficient of uv, which means that, the decay of the prey decreases after we added the term uv. Also in the predator equation we have added the term uv with rate a 8, and a 8 > a 5, which means that, the growth rate of the predator increases after we add the term uv. Therefore the interaction between the predator and the prey with the term uv is better than the interaction with the term uv. B- The interaction between two predators and one prey is possible even though we are at closed environment that is for several reasons: )i) The temperature has an effect on movement of the predator and prey. For example if it increases, the predator's movement will slow down. And if it decreases the prey will hide. Consequently, two predators will gather on a prey.
8 676 M. A. Al Qudah (ii) In the mating period we can consider the term v is the existence of a male and a female together, this will lead two predators to meet on a prey. As the predator has a strong rapacity and can take more than his needs so it is better to consider the term uv. In both cases, low temperature and mating, the consumption of food will decrease and providing food will be continuous to all the species. References [] M. S. Abualrub, Diffusion Problems in Mathematical Biology, Dirasat Natural and Engineering Sciences, 3 (996), no., 6-5. [] M. S. Abualrub, Long Range Diffusion-Reaction Model on Population Dynamics, Documenta Mathematica, 3 (998), [3] M. S. Abualrub, Traveling Wave Solutions to Predator-Prey and Insect Dispersal Models, Accepted for Publication in the Journal of Franklin Institute, USA. (5). [4] M. S. Abualrub, Existence and Uniqueness of Solutions to a Diffusive Predator-Prey Model, Journal of Applied Functional Analysis, 4 (9), no., 7-. [5] M. A. Al-Qudah, and M. S. Abualrub, Existence of Solutions to a Model of Long Range Diffusion Involving Flux, Journal of Applied Functional Analysis, 5 (), no. 4, [6] M. A. Al-Qudah, Stability of Solutions for Some Non-Linear Partial Differential Equations, PhD.Thesis, University of Jordan,. [7] M. A. Al-Qudah, and M. S. Abualrub, Solitary and traveling wave solutions to a model of long range diffusion involving flux with stability analysis, International Mathematical Forum, 6 (), no. 7, [8] A. Benedek and R. Panzone, The space L^pwith mixed norm, Duke Math. J., 8 (96), [9] C.P. Clarderon, Diffusion and nonlinear population Theory, Rev. U. Mat., 35 (99), [] P. C. Fife, Stationary Patterns for Reaction-Diffusion Equations, Pitman Research Notes in Mathematics, Eds, W. E. Fitzgibbon, III and H. W. Walker, 4 (977), 8-.
9 Existence, uniqueness solution of a modified predator-prey model 677 [] S.A. Levin, and L.A. Segel, Hypothesis for the Origin of Planktonic Patchiness, Nature, 59 (976), [] A. J. Lotka, Elements of Mathematical Biology, New York Dover Pub, 956. Received: September 8, 6; Published: November 5, 6
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