Long Range Diffusion Reaction Model on Population Dynamics

A model for long range diiusion reaction on population dynamics has been considered, and conditions for the existence and uniqueness of solutions to the model in L p;q norms has been obtained.


Introduction
The dynamics of population has been described using mathematical models which have been very successful in giving good e ect in the study of animal and human populations.Fisher 4] introduced a model for the spatial distribution of an advantageous gene as non-linear di usion equations.Later, Hoppensteadt 6] p.50, derived an equation of age-dependent population growth which involves rst order partial derivatives with respect to age and time, where Fife 3] considered reaction and di usion systems which are distributed in 3-dimensional space or on a surface rather than on the line.In addition, Abual-rub studied di usion in two dimensional spaces for which di usion is more realistic and applicable in life.Most of these di usion models deal with usual di usion or short range di usion.Such models have played a major role in the study of population dynamics.However, long range di usion could also have a big in uence on the dynamics of some populations with the form it takes depending on the nature of the populations themselves.Abual-rub talked about long range diffusion with population pressure in Plankton-Herbivore populations.He considered a model of the following form: P t c (2) P = aP + eP 2 bPH + u + 1 P +1 (1) P(x; 0) = g(x); x 2 R 2 ; (2) and H t ` (2) H(x; 0) = h(x); x 2 R 2 ; (4) where P(x; t) and H(x; t) represent the Plankton and Herbivore densities, respectively.
Here represents the Laplacian operator and (2) = 2 X i;j=1 @ 4 @x 2 i @x 2 j : ( The existence and uniqueness of solutions to (1)-( 4) have been proved by Abualrub in the L p;q spaces.Okubo 8] p. 194, discussed the e ect of density-dependent dispersal on population dynamics by considering the Gurtin and MacCamy 5] model which combines the ux with the population reaction term, F(S), he considered di usion-reaction problems in one dimension of the form: @S @t = K @ 2 S m+1 @x 2 + F(s); (6) where K = k (m + 1) > 0: (7) Murray 7] p.245, which is one of the good books in mathematical biology, considered a long range di usion model of population by taking the ux J to be: where D 1 and D 2 are the constants which measure short range and long range e ects, respectively.He obtained a long range di usion approximation of the form: @S @t = r D 1 rS r r(D 2 S): (9) For this model, Murray mentioned that the e ect of short range di usion is, usually, larger than that of long range di usion, i.e.D 1 > D 2 .In this paper we will see what happens if the e ect of long range di usion is larger.This assumption might not be realistic in general, but we think that it might be true in some rare cases of population dynamics such as for certain epidemics and Plankton-Herbivore systems.
We will consider the two dimensional case in our model rather than the rst dimensional case i.e., x = (x 1 ; x 2 ); because it is more realistic that di usion takes place in spaces and not along lines.Therefore, we will use S instead of @2 S @x 2 .As mentioned in the introduction we will assume that the e ect of long range di usion is larger than that of short range di usion and investigate what will happen if at some stage D 1 is negligible compared with D 2 .We believe that this might happen at some stages depending on the nature of the population and the nature of its dynamic.Its known that in short rang di usion the ux J takes the following form J = DrS: (10) Murray 7] p.245, derived the equation for ux J in (8).In our model, according to the above assumptions, we will consider the ux to be of the form J = r (D 2 S) : (11) The conservation equation for S is given by @S @t = r J + F(S); (12) where F(S) is the population reaction term.By substituting (11) into (12) we get the following model for long range di usion reaction, namely @S @t = D 2 (2) S + F(S) (13) In this paper we will impose the initial condition on S , namely S(x; 0) = g(x) (14) In addition, we will consider F(S) to be directly proportional to S n , i.e, F(S) = aS n (15) for some positive constant a and integer n which has to be determined later.The reason for writing S n here is that in usual di usion we have always S or S 2 but in long range di usion things might di er and if it does we want to determine the right exponent, n, for S. Let C = D 2 , our model is thus @S @t C (2) S = aS n ; (16) S(x; 0) = g(x): (17) where the term C (2) S represents long range di usion.
3 Existence and uniqueness of solutions: We will look for solutions to model ( 16), (17) in the L p;q space, the function space consisting of Lebesgue measurable functions S(x; t) such that kSk p;q < 1, where k( )k p;q is the norm in L p;q de ned by : kSk p;q = " Z T 0 Z R 2 jSj p dx q p dt # 1 q (18) We will now state and prove the main result in this paper.

LEMMA
The solution to model ( 16), (17), S(x; t), exists and is unique in the space L 3 2 (n 1); 1 2 (n 1) for n > 3, whenever the initial data g (x) is small enough in the norm of its space.Proof.We begin by transforming equation ( 16) and the initial condition (17) into the following integral equation We will now rewrite (19) simply as S = aK S n + K g (20) where denotes the convolution in space and time and denotes the convolution in space only.Here the kernel K is the Fundamental solution to the homogeneous problem of (16), namely K(x; t) = t where is a constant.We now take the q norm in x of the above inequality to obtain kK gk p;q c Z R 2 jg(y)j dy jx yj + t 1=4 2 4   p q : The right hand side of the above inequality is less than or equal to constant kgk q ; if 1 p = 1 q 4 2p (using the Benedek-Panzone Potential Theorem 1], see Appendix).This implies that p = 3q and hence K g 2 L 3q To get a contraction mapping (see appendix) L p R 2 R + !L p R 2 R + in (20), the exponents in ( 23) and (26) must be equal, that is Now, its enough to show the uniqueness of the solution.
Lets apply the mapping T to (20) to obtain : T(S) = aK S n + K g (32) Its easy to see that: where h is an auxiliary function which represents the term K g in (32).
We are going now to compare equation (33) to the following mapping : where both and are positive constants.Of course x n is convex and increases faster that a linear function.
Its obvious to see that if = 0, there is only one non-zero root of (34) but if 0 < < (where is su ciently small), we will have two roots, say f x 1 and f x 2 : Let f x 1 be the smallest root, then if f x 1 is small enough then the mapping T will be a contraction mapping which maps the ball of radius f x 1 into itself.This implies that the solution to the equation S = T(S) in (32) exists and its unique in the ball of radius f x 1 .This concludes the proof of Lemma 3.1.
Remark 1: The extension of the results in Lemma 3.1 to three or n dimensions is straight forward.Remark 2: See 2] for a general method for studying long-time asymptotics of nonlinear parabolic partial di erential equations.In 2], p.898, Remark 1, the existence and uniqueness of solutions have been shown.Comparing our results with the results obtained in 2], we conclude that if we take = 4, then equation (8) in 2], p. 898, is analogous to our equation ( 16) here and u(x; t) used in 2] is the same as K(x; t) used here in (21).This shows that our method coinsides with the method used in 2] and thus therorem 1 in 2] is applicable to our case.Documenta Mathematica 3 (1998) 333{340 concludes the proof for the initial data.Now, for the rst term in (20), note that we can rewrite (calculations to the rst term in (20), using (24), similar to what has been done to the second term in (20) in the previous page 5, using (22), then applying the Benedek-Panzone Potential Theorem 1], see appendix, we conclude that 1 setting r = p in (25) we get :