Kolade M. Owolabi Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein 9300, South Africa e-mails:
[email protected];
[email protected]
Abdon Atangana Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein 9300, South Africa
Numerical Simulation of Noninteger Order System in Subdiffusive, Diffusive, and Superdiffusive Scenarios In this work, we investigate both the mathematical and numerical studies of the fractional reaction–diffusion system consisting of spatial interactions of three components’ species. Our main result is based on the analysis of the model for linear stability. Mathematical analysis of the main equation shows that the dynamical system is both locally and globally asymptotically stable. We further propose a theorem which guarantees the existence and permanence of the three species. We formulate a viable numerical methods in space and time. By adopting the Fourier spectral approach to discretize in space, the issue of stiffness associated with the fractional-order spatial derivatives in such system is removed. The resulting system of ordinary differential equations (ODEs) is advanced with the exponential time-differencing method of ADAMS-type. The complexity of the dynamics in the system which we discussed theoretically are numerically presented through some numerical simulations in 1D, 2D, and 3D to address the points and queries that may naturally arise. [DOI: 10.1115/1.4035195] Keywords: asymptotically stable, coexistence, exponential time-differencing scheme, Fourier spectral method, numerical simulations, predator–prey, fractional reaction– diffusion systems
1
Introduction
Over the years, reaction–diffusion systems emanated from the study of multispecies Lotka–Volterra interactions (namely; predator–prey, competition, mutuality, and food-chain) have been the subject of activities [1,2]. Recently, many authors studied three-species population dynamics with functional responses, impulsive effects, time delays, and stage-structure (see, for instance, Refs. [3,4]) and obtained some results on permanence, global existence of solution, asymptotic stability, stability or instability of the equilibrium states, and periodicity. In this research work, we give an extension to the study of population dynamics from two species predator–prey model to a three-species reaction–diffusion systems consisting of two preys and one predator with impulsive effect. Many ecological processes are governed by the fact that they experience a sudden change of state at certain moment of time. These processes arise as a result of short-time perturbation with a very small time-lag in comparison with the period of the process. In natural sense, it is reasonable to assume that perturbation arises in the form of impulse, and most biological phenomena involving pharmacokinetics systems, thresholds, and optimal control models exhibit some kind of impulsive effects. Various works have been done where impulsive control strategy is used to study predator–prey dynamics, see Ref. [5] and the references therein. A coupled fractional reaction–diffusion system of n ðn 3; n integerÞ species which interacts in a nonlinear fashion and diffuse may be modeled by the equations
Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received May 3, 2016; final manuscript received October 28, 2016; published online December 5, 2016. Assoc. Editor: Dumitru Baleanu.
ut D1 Dg=2 u ¼ f ðu; v; wÞ vt D2 Dg=2 v ¼ gðu; v; wÞ in X ½0; 1Þ; t 6¼ TðÞ wt D3 Dg=2 w ¼ hðu; v; wÞ B½u ¼ B½v ¼ 0; B½w ¼ S on @X ½0; 1Þ; t ¼ TðÞ uðx; 0Þ ¼ u0 ðxÞ; vðx; 0Þ ¼ v0 ðxÞ; uðx; 0Þ ¼ u0 ðxÞ; ðx; tÞ 2 X (1) where D is the Laplacian operator in one, two, or more dimensional space, Dg=2 is the Riemann–Liouville fractional integration of order g given as ðx g 1 ð x nÞ21 F ðnÞdn; g > 0; x > 0; g 2 0 < g 2 Dg=2 ¼ Cðg=2Þ 0 which we classify into subdiffusive for 0 < g < 1, diffusive for g ¼ 2, and superdiffusive in the open interval 1 < g < 2. The reaction terms are given as u a1 u1 uw f ðu; v; wÞ ¼ s1 u 1 j1 b1 þ u þ c1 w v a2 u2 vw gðu; v; wÞ ¼ s2 v 1 j2 b2 þ v þ c2 w q1 a1 u hðu; v; wÞ ¼ w1 u1 w b1 þ u þ c 1 w q2 a2 v þ w2 u2 w b2 þ v þ c2 w
Journal of Computational and Nonlinear Dynamics C 2017 by ASME Copyright V
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MAY 2017, Vol. 12 / 031010-1
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The X is a bounded region in R3þ with boundary @X. The diffusion coefficients Di ði ¼ 1; 2; 3Þ, the intrinsic growth rates si ði ¼ 1; 2Þ, and the cropping rate ai ði ¼ 1; 2Þ are all positive parameters, the carrying capacity of the preys (u, v) ji, the saturation constants bi, the predator interference ci, predator death rates wi in patch i, and ui which represents the segment of lifetime an average predator waits in patch i, the rate at which resources are converted to a new consumers is denoted by qi, for i ¼ 1, 2 are all nonnegative constants, and we also assume that the initial functions u0, v0, and w0 are all nonnegative constants. Period of the impulsive effect is denoted by T, 2 N, N is a set of positive integers [5], S > 0 is the amount release by predator w at t ¼ TðÞ. The idea of fractional calculus is seen as a subject of considerable interest in the fields of physics, mathematics, and engineering [6,7]. In recent years, the study of fractional derivatives has gained a significant development in both ordinary and partial differential equations. A lot of physical problems are modeled mathematically with systems of fractional differential equations. Finding efficient and accurate methods to numerically simulate fractional differential equations is an active research undertaken in this paper. Among many authors that have studied the numerical simulations of such problems can be found in Refs. [8–12] The objectives of the present paper are in folds: We first proof the conditions that guarantee that system (1) is both locally and globally asymptotically stable for the preys (u, v) eradication periodic solutions. Second, since we can split Eq. (1) into reaction and diffusion terms, the use of two classic mathematical ideas is introduced. Hence, we are to formulate viable adaptive methods in space and time for the numerical simulations [12]. We design a versatile Fourier spectral method in space which to be used in conjunction with an exponential time-differencing scheme for the time integration. We also report the results of some experiments with ecological and numerical implications.
2
Mathematical Analysis of the Main Results
In the spirit of Ref. [13], we examine the local stability analysis of Eq. (1). Let Rþ ¼ ½0; 1Þ; R3þ ¼ fX 2 R3 jX 0g. We let G ¼ ðf ðu; v; wÞ; gðu; v; wÞ; hðu; v; wÞÞT as the map that defines the reaction terms. Let H : Rþ R3þ ! Rþ , then H 2 H0 if; H is continuous in ðTðÞ; Tð þ 1Þ R4þ , for each X 2 R4þ ; 2 N; limðt;ZÞ!ðT þ ðÞ;XÞ Hðt; ZÞ ¼ HðT þ ðÞ; XÞ exists. And, H is said to be locally Lipschitzian in X. The local stability analysis is examined in order to see behavior of the local dynamics in the absence of diffusive terms, that is, when Di ¼ 0; i ¼ 1; 2; 3. The steady-states of system (1) are determined by setting the reaction terms f ðu; v; wÞ ¼ gðu; v; wÞ ¼ hðu; v; wÞ ¼ 0. DEFINITION 2.1. Suppose that function GðtÞ ¼ ðuðtÞ; vðtÞ; wðtÞÞ is smooth and satisfies Eq. (1) in R3þ , and every component of G is almost periodic function, we say that G is a spatial homogeneity periodic solution of Eq. (1) represented by Gðt; TðÞÞ, for every > 0. THEOREM 2.2. Assume (u, v, w) to be any solution of model (1), ^ which corresponds to the then the equilibrium point ð0; 0; wÞ eradication of preys is locally asymptotically stable if a1 u1 c1 ðw1 u1 þ w2 u2 Þ c1 S ð1 expðw1 u1 T w2 u2 T ÞÞ ln 1 þ s1 T < 0 c1 S þ b1 ð1 expðw1 u1 T w2 u2 T ÞÞ (3) and a2 u2 c2 ðw1 u1 þ w2 u2 Þ c2 S ð1 expðw1 u1 T w2 u2 T ÞÞ ln 1 þ s2 T < 0 c1 S þ b2 ð1 expðw1 u1 T w2 u2 T ÞÞ (4) 031010-2 / Vol. 12, MAY 2017
Proof. We can determine the local stability of prey eradication ^ by examining the behavior of system periodic solution ð0; 0; wÞ (1) when subjected to a small amplitude perturbation þ wÞ ^ ðu; v; wÞ ¼ ð u ; v; w
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Using Eq. (5) in Eq. (1), we have the linearized system written in the form ^ d u a 1 u1 w ¼ u s1 ^ dt b1 þ c1 w ^ d v a 2 u2 w ¼ v s2 t 6¼ T ðÞ (6) ^ dt b2 þ c2 w dw ðw1 u1 w2 u2 Þ ¼w dt ¼ 0; t ¼ T ðÞ B½u ¼ B½v ¼ B½w which leads to T u ð0Þ; vð0Þ; wð0ÞÞ ; ð u ; v; uÞT ¼ Að
0tT
where A at point ð u ð0Þ; vð0Þ; wð0ÞÞ satisfies 0 1 0 0 s a dA @ 1 0 s2 b 0 AAðtÞ ¼ dt 0 0 c
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^ ^ b ¼ a2 u2 w=ðb ^ ^ c ¼ w1 u1 with a ¼ a1 u1 w=ðb 1 þ c1 wÞ; 2 þ c2 wÞ; þ ðÞ þ w2 u2 ; Að0Þ ¼ I; is the identity matrix; and ½ð u ; v; wÞT ¼ ½1; 0; 0; 0; 1; 0; 0; 0; 1½ð u ; v; wÞTðÞ: So, stability of the periodic prey eradication at solution ^ can be examined by the eigenvalues of point ð0; 0; wÞ ½1; 0; 0; 0; 1; 0; 0; 0; 1AðTÞ which have an absolute value of less ^ is locally stable. Based on than one. Then, the solution ð0; 0; wÞ Floquet theory, all the eigenvalues ! ! ð ð T
k1 ¼ exp
T
ðs1 aÞ
0
k2 ¼ exp
ðs2 bÞ
0
k3 ¼ expðcðTÞÞ < 1 ^ is locally asymptotically stable if Hence, the point ð0; 0; wÞ jk1 j < 1 and jk2 j < 1, which imply that conditions (3) and (4) are satisfied. The proof is completed. 䊏 To keep the problem open for other researchers to explore, we like to present the conditions for system (1) to be globally asymptotically stable and permanent via a theorem. THEOREM 2.3. Given (u, v, w) to be any solution of model (1), ^ which corresponds to the prey eradication then the point ð0; 0; wÞ is globally asymptotically stable if a1 u1 c1 ðw1 u1 þ w2 u2 Þ c1 S ð1 expðw1 u1 T w2 u2 T ÞÞ ln 1 þ s1 T < 0 c1 S þ b1 ð1 expðw1 u1 T w2 u2 T ÞÞ a2 u2 c2 ðw1 u1 þ w2 u2 Þ c2 S ð1 expðw1 u1 T w2 u2 T ÞÞ ln 1 þ s2 T < 0 c1 S þ b2 ð1 expðw1 u1 T w2 u2 T ÞÞ and
c ðw u þ w2 u2 ÞT ðb1 þ J þ c1 J Þ S max 1 1 1 ; a1 u1 c2 ðw1 u1 þ w2 u2 ÞT ðb2 þ J þ c2 J Þ a 2 u2 Transactions of the ASME
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For a nonnegative constant J , we have u J ; v J ; w J for each solution X ¼ ðu; v; wÞ of ecological model (1), for t large. Again, system (1) is said to be permanent if a1 u1 c1 ðw1 u1 þ w2 u2 Þ c1 S ð1 expðw1 u1 T w2 u2 T ÞÞ ln 1 þ s1 T > 0 c1 S þ b1 ð1 expðw1 u1 T w2 u2 T ÞÞ a2 u2 c2 ðw1 u1 þ w2 u2 Þ c2 S ð1 expðw1 u1 T w2 u2 T ÞÞ ln 1 þ s2 T > 0 c1 S þ b2 ð1 expðw1 u1 T w2 u2 T ÞÞ and
as H ! 0, so also u; v ! 0 for t ! 1. Hence, we conclude that the system (1) is globally asymptotically stable for the prey^ extinction solution ð0; 0; wÞ. The proof of second part of Theorem 2.3 is omitted in this paper, details result similar to it can be found in the book [13].
3 Formulation of Adaptive Methods in Space and Time Spectral methods have been considered as the logical extension of conventional finite differences to infinite order [14,15], because of its ability to remove the stiffness property that is associated with the linear term of a reaction–diffusion problems [16]. Based on the known integrating factor technique, we shall formulate the theory of spectral method in one spatial dimension. In a compact form, system (1) can be written as
c ðw u þ w2 u2 ÞT ðb1 þ J þ c1 J Þ S < max 1 1 1 ; a1 u1 c2 ðw1 u1 þ w2 u2 ÞT ðb2 þ J þ c2 J Þ a 2 u2
Proof. Assume H ¼ j1 u þ j2 v, from Eq. (1) in absence of diffusion, we obtain a1 j1 u1 uw a2 j2 u2 vw þ j2 s2 v s2 v2 H ¼ j1 s1 u s1 u u þ c1 w þ b1 v þ c2 w þ b2 0
For t > t1 , we have a1 j1 u1 C a 2 j 2 u2 C u þ j2 s 2 v 0, and a small positive constant, ^ d, valid for all t t1 . We have say d, in such a way that w w S expððw1 u1 þ w2 u2 ÞT Þ w d 1 expððw1 u1 þ w2 u2 ÞT Þ and H0
j1 s1
a1 j1 u1 C a2 j2 u2 C u þ j2 s2 v Q þ c1 Q þ b1 Q þ c2 Q þ b2
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g=2
wt ¼ D3 D w þ hðu; v; wÞ |fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} L
Also, we let a constant Q > 0 in such a way that u; v; w Q for any solution (u, v, w) of system (1) with all t > 0 and large enough. Then,
0
