The Existence Of Positive Periodic Solutions For Biological Systems

Based on many references, in this paper we summrize the problem for the existence of periodic solution for biological systems using the coincidence degree method. Although these biological models are different kinds of differential equations which are nonautonomous differential equations, difference equations and impulsive differential equations respectively, using same coincidence degree theorem, we can get the existence of periodic solutions for three different biological systems which verify this functional theory is universal for nonlinear biological system whatever its dimension.In order to attack the existence of positive periodic solutions, for the reader's convenience, we shall recall now some basic tools in the frame of Mawhin's coincidence degree that will be used to establish sufficient criteria for the existence of positive periodic solutions of biological systems.The important concepts and theorem as follows:Let X, Y be normed vector spaces, L : DomL (?) X → Y be a linear mapping, N : X → Y be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dimKerL = codimlmL < +∞ and ImL is closed in Y. If L is a Fredholm mapping of index zero and there exist continuous projectors P : X→X and Q :Y → Y such that ImP = KerL, ImL = KerQ =Im{I - Q), it follows that L\DomL n KerP : (/ - P)X -? ImL is invertible. We denote the inverse of that map by Kp. If it is an open bounded subset of X, the mapping N will be called L-compact on fl if QN(Q) is bounded and Kp(I - Q)N : fj —? X is compact. Since 7mQ is isomorphic to KerL, there exists an isomorphism J : ImQ —? KerL.Lemma 2.1 (Continuation Theorem) Let L be a Predholm mapping of index zero and N be L-compact on Cl. Suppose(a). For each A (E (0,1), every solution x of Lx = XNx is such that x 0 d￡l;(b). QNx ^ 0 for each xediln KerL anddeg{JQN, Q n KerL, 0} ^ 0.Then the operator equation Lx = Nx has at least one solution lying in DomLnCl.The Key problem for applying to it is constructing a proper set which satisfies all conditions in the theorem.In this paper we consider three different biological system which is made of different type differential equations, that are follows:I. predator-prey system of three species:*, -where b^aij^c^mj 6 C([0,oo), (0,oo)), (i = 1,2,3;j = 1,2) are ^-periodic functions. (1) is a food-chain model of three species: X\ is preyed by x%, meanwhile^ is predator of ￡3.Let§ = - fg(t)dt, G = - [\g(t)\dt, 0 0= min {yi(t)}, 2/1(571) = max{yi(t)}, (￡*, 77^ € [0,w]), m" = max{m(f)}.For this model it can get the following results:Theorem 2.1.3 Suppose Si - ( — ) > 0, /i - S2 - ( — ) > 0, and /2 -Vmi/ \m2ya3 > 0, then there is at least a u;-periodic solution for (1) which is y*(t) — > where there is a positive constant M such that ||y*|| < M,3Hl/11 = (y>;io)1/2. brio = max|yr(fc)|(i = 1,2,3).II. n—species competitive system:Vi(k + 1) = Vi{k) exp I r^fc) - ￡-E E cy(*??)y^(?)-i:j=lu=-oo j=lSuppose:Ul-l(Hi) r?, py, hij : Z —? R are u> periodic functions such that 2~2 r?(s) > 0;o(H2) ajj, bij, dij : Z —> R+ are nonnegativewperiodic functions;(H3) cij:ZxZ^R+ satisfy cyCfc+w,s + w) = ^(fc,t)| J] cy(ik,s) <+oo;(H4) a^, (3^, jij are positive constant, u> is given integer denoting the coefficients' period of system(2).System(2)means a n—species competition models which is not overlay ofevery age, it can take a discrete similarity of continuous equations which read= Vi(t)i(t) - E aijWvT'3=1n +00 n- E / ci:i(t,U)y]"(t - u)du - E dlj(t)yt(t)yj(t - C,tj j=\ 0 j=ii = l,2,---,n,This is a universal biological system which include many simple kinds such as (1) Lotka-Volterra competitive system with infinite delay;(2) Gilpin-Ayala competitive system;(3) Plankton Allelopathy system.Theorem 2.2.3 Suppose (Hi)-(H4) holds and (H5) algebric equationf(u) := U - J2 (a??r + M?' + *iuV + *nxlhave finite solution u* in ZniR" and Es^3n//(u*) 7^ 0;Furthermore, ifu'an - Ai = 7?;r4 > J>yA^ + ^Af + gyAf) + ^ 4A^whereexp{(f'w-i itk=0 s=-oothen (2)have at least a aperiodic solutiony*(￡) and there exists positive constant satisfying a*, A > 0 such that a* < y*(t) < /?,.III. Two-species cooperative system:&yi(t)\t=tk = vi(tt) - yi(t;) = 61*1/1 (tit);- c2(t)y2(t)(3)fcl fc = 1,2,-?? .For system(3), suppose:(ffi) 6j/t > 0 ^0 &2fe > 0 denote population y\ and y2 impulsive birth rates at time tk;(#2) Vi(tt) an<^ J/?(*fc) mean left-hand limit and right-hand limit of y, at time tk , suppose y^ is left continuous at tk ■(H3) All the coefficients functions are continuous nonnegative u periodic functions;(#4) there exists positive integer q such that tk+q = tk + u>, b^k+q) — kk, suppose tk ^ 0 and [0,u;] n {tk} = {ti,t2, ■ ■ ? ,tm}, so we can q = m.Theorem 2.3.3 Under the conditions (Hi)-(H4), system (3) exists at least a positive aperiodic solutions.