DYNAMICS IN 3-SPECIES PREDATOR-PREY MODELS WITH TIME DELAYS. Wei Feng

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1 DISCRETE AND CONTINUOUS Website: DYNAMICAL SYSTEMS SUPPLEMENT 7 pp DYNAMICS IN 3-SPECIES PREDATOR-PREY MODELS WITH TIME DELAYS Wei Feng Mathematics and Statistics Department University of North Carolina at Wilmington Wilmington, NC83-597, USA Abstract. We study a differential equation system with diffusion and time delays which models the dynamics of predator-prey interactions within three biological species. Our main focus is on the persistence (non-extinction) of u-species which is at the bottom of the nutrient hierarchy, and the permanence effect (long-term survival of all the predators and prey) in this model. When u-species persists in the absence of its predators, we generate a condition on the interaction rates to ensure that it does not go extinction under the predation of the v- and w-species. With certain additional conditions, we can further obtain the permanence effect (long-term survival of all three species) in the ecological system. Our proven results also explicitly present the effects of all the environmental data (growth rates and interaction rates) on the ultimate bounds of the three biological species. Numerical simulations of the model are also given to demonstrate the pattern of dynamics (extinction, persistence, and permanence)in the ecological model. 1. Introduction. A simple and realistic model for single species population dynamics (with the assumption that the growth rate decreases linearly according to the population size) is the famous Logistic Equation [9] X (t) = rx(t)[1 X(t)/K], X() = X ; (1) where K is the so-called carrying capacity (or the maximum sustainable population)for the biological species. We may further think of a single species grazing upon vegetation, which takes time τ to recover. In this case, the population size with time delay X(t τ) will directly affect the growth rate of the population at time t. The logistic delay differential equation X (t) = ax(t)[1 X(t τ)/k] () has also been the object of intensive analysis by numerous authors (see, for example, [9]). It is known [1](Wiener and Cooke, 1989)that time delay has a tendency to produce oscillations in equation (3) which is non-oscillatory when τ =. In recent years, researchers also paid attention to the boundedness of the habitat and the diffusion effect of the species within it. In 1997, Feng and Lu [] studied the following diffusive logistic equation with instantaneous and discrete delay effects u t = D[ u + u(a αu βu τ in Ω (, ), u = on Ω (, ), (3) Mathematics Subject Classification. Primary: 58F15, 58F17; Secondary: 53C35. Key words and phrases. Mathematical Ecology, Differential equation models, Predator-prey systems, Time delays, Long-term survival and permanence. 36

2 3-SPECIES MODELS WITH DELAYS 365 u(x, s) = u (x, s) in Ω [ τ, ], and obtained results on the asymptotic limits of the solution u (to be given in the Preliminary section). In this paper we extend the study of population dynamics from single species models to a model of three species predator-prey interaction [1, 3,, 6] with diffusion and delays. In an ecological system with three species: A, B and C where B and C both consume A. let u(x, t), v(x, t) and w(x, t) be the corresponding scaled density functions of A, B and C. We also would like to consider the diffusion effect and time delay effects in the interactions of the species, caused by the time period of food digesting and recovery of the preys. This leads to a differential equation system with time delays modeling the Lokta-Volterra type of interactions among the species [1]: u t = D 1 [ u + u(a 1 u l 11 u τ 11 k 1 v l 1 v τ 1 k 13 w l 13 w τ 13 v t = D [ v + v(a + k 1 u + l 1 u τ 1 v l v τ k 3 w l 3 w τ 3 w t = D 3 [ w + w(a 3 + k 31 u + l 31 u τ 31 k 3 v l 3 v τ 3 w l 33 w τ 33 in Ω (, ) () u = v = w = on Ω (, ) u(x, s) = u (x, s), v(s, x) = v (x, s), w(s, x) = w (x, s) in Ω [ τ, ], where Ω is a bounded region in R 3 with smooth boundary Ω. The diffusion term u is represented by the Laplacian Operator, and the diffusion coefficients D i and the growth rates a i (i = 1,, 3) are all positive constants. As in many real-life food webs, we assume that a 1 a > and a 1 a 3 >, namely, the species higher in the trophic levels have relatively smaller growth rates. The interspecific interaction rates k ij and l ij (1 i, j 3) are all nonnegative constants. Through appropriate scaling of the density functions, we can obtain that k 11 = k = k 33 = 1, for the convenience of our later discussions. The initial functions u, v, and w are assumed to be nonnegative smooth functions in [ τ, ]. In system () we also denote the delay terms u τij = u(x, t τ ij ), v τij = v(x, t τ ij ), and w τij = w(x, t τ ij ), with τ = max{τ ij, 1 i, j 3}. The main objective of this paper is to answer the following important questions in population dynamics concerning model (): Determine the existence and boundedness of the density functions (u, v, w) for all three of the species in [, ) Ω. Find the conditions for the persistence (long-term survival) of the prey species(u) at the bottom of the trophic hierarchy. Establish a permanence criterion (long-term survival of all three species) in the ecological system (). Find the the impact of the environmental data on the behavior of the biological species and their ultimate bounds.. Preliminaries. In the absence of the two predators, the prey species u satisfies the following single species model with time delays u t = D[ u + u(a αu βu τ in Ω (, ), u = on Ω (, ), (5) u(x, s) = u (x, s) in Ω [ τ, ],

3 366 WEI FENG where u τ (x, t) = u(x, t τ with τ >. The diffusion rate D, the scaled growth rate a, the interaction rates α and β are all positive constants, with α + β = 1. Denote by λ > the smallest eigenvalue of the eigenvalue problem: φ + λφ = in Ω, φ = on Ω. (6) Feng and Lu [] obtained the following result concerning the solutions of the Logisticdelay system () and its corresponding steady-state problem φ + φ(a φ) = in Ω, φ = on Ω. (7) Lemma 1. (i) If a λ, then the boundary value problem (7) has only the trivial solution and the solution u(x, t) of (5) converges to uniformly on Ω as t. (ii) If a > λ and α > β, then the boundary value problem (7) has a unique positive solution Φ(x). For every nonnegative and nontrivial u, the solution u(x, t) of (5) converges to Φ(x) uniformly on Ω as t. The question on existence and global boundedness of the non-negative density functions in the 3-species predator-prey model () can be dealt with by applying the theory of upper-lower solutions given by Pao in 199 [7]. Lemma. If there exist a pair of smooth functions (ũ, ṽ, w) and (û, ˆv, ŵ) such that (ũ, ṽ, w) (û, ˆv, ŵ) on [, ) Ω and they satisfy the following inequalities: ũ t D 1 [ ũ + ũ(a 1 ũ l 11 û τ11 k 1ˆv l 1ˆv τ1 k 13 ŵ l 13 ŵ τ13 ṽ t D [ ṽ + ṽ(a + k 1 ũ + l 1 ũ τ 1 ṽ l 1ˆv τ k 3 ŵ l 3 ŵ τ 3 w t D 3 [ w D 3 w(a 3 + k 31 ũ + l 31 ũ τ31 k 3ˆv + l 3ˆv τ3 w l 33 ŵ τ33 û t D 1 [ û D 1 û(a 1 û l 11 ũ τ 11 k 1 ṽ l 1 ṽ τ 1 k 13 w l 13 w τ 13 ˆv t D [ ˆv + ˆv(a + k 1 û + l 1 û τ 1 ˆv l 1 ũ τ 1 k 3 w l 3 w τ 3 ŵ t D 3 [ ŵ + ŵ(a 3 + k 31 û + l 31 û τ31 k 3 ṽ l 3 ṽ τ3 k 33 ŵ l 33 w τ33 (ũ, ṽ, w) (,, ) (û, ˆv, ŵ) on Ω (, ) (ũ, ṽ, w) (u, v, w ) (û, ˆv, ŵ) in Ω ( τ, ), then there exists a unique solution (u, v, w) for () on [, ) Ω with (ũ, ṽ, w) (u, v, w) (û, ˆv, ŵ). We let M 1 = max{ a 1, u }, M = max{ a + k 1 M 1, v }, M 3 = max{ a 3 + k 31 M 1, w }. It is not hard to verify that (M 1, M, M 3 ) and (,, ) satisfy all the requirements given in Lemma for coupled upper and lower solutions of (). Our result on the existence and global boundedness for the non-negative density functions is given as follows: Theorem 1. The system of differential equations with time delays () has a unique solution (u, v, w) on Ω [, ) with (,, ) (u, v, w) (M 1, M, M 3 ).

4 3-SPECIES MODELS WITH DELAYS Our Main Results: Persistence and Permanence. As seen in Section, a previous result on diffusive logistic equation with time delays shows that in the absence of its predators v and w in (), the u-species at the bottom of the nutrient hierarchy will persist when a 1 > λ and l 11 < 1. Through a series of comparison analysis on the three differential equations in system (), we will show in this section that when a 1 > λ and δ = l 11 + (k 1 + l 1 )(1 + k 1 + l 1 ) + (k 13 + l 13 )(1 + k 31 + l 31 ) < 1, the u-species at the bottom of the trophic hierarchy will persist. This result involves the growth rate a 1 of u-species and all the interaction coefficients (k ij and l ij ) in (). Furthermore, we also obtain the ultimate lower bound of the density function u in term of the interaction rates by showing that lim inf u(, t) (1 δ)φ 1 ( ). Theorem. Persistence of the u-species. Let a 1 > λ and Φ 1 (x) be the unique positive solution of the boundary-value problem (7) when a = a 1. If δ = l 11 + (k 1 + l 1 )(1 + k 1 + l 1 ) + (k 13 + l 13 )(1 + k 31 + l 31 ) < 1, (8) then the density function u satisfies lim inf u(, t) (1 δ)φ 1( ) in C(Ω). (9) Proof: 1. Let (u, v, w) be the time-dependent solution of (). From the fact that u, v, and w are all non-negative functions, we know that u satisfies u t D 1 [ u + u(a 1 u in Ω (, ). Hence u is a lower solution for equation (5) in Section when β =. Lemma 1 indicates that the solution for equation (5) converges to Φ 1 (x) when t. Therefore u satisfies lim sup u(, t) Φ 1 ( ) in C(Ω).. Using the same argument as above and the ultimate upper bound Φ 1 for u, we know that for any arbitrarily small ɛ >, there exists a T ɛ >, v t D [ v + v(a + (k 1 + l 1 + ɛ)φ 1 v D [ v + v(a 1 + ɛ + (k 1 + l 1 )Φ 1 v in Ω (T ɛ, ). Noting that (1 + k 1 + l 1 )Φ 1 (x) is the solution of the boundary value problem V + V [a 1 + (k 1 + l 1 )Φ 1 V ] = in Ω, V = on Ω, we can conclude that the density function v satisfies: lim sup v(, t) (1 + k 1 + l 1 )Φ 1 ( ) in C(Ω). For the same reason, we can also obtain lim sup w(, t) (1 + k 31 + l 31 )Φ 1 ( ) in C(Ω). 3. Finally we apply the results in part 1 and part to the first equation in () for u. After substituting the ultimate upper bounds for u, v, w into the equation, we can obtain that for arbitrary ɛ >,there exists a T ɛ > such that u t D 1 [ u + u[a 1 u (δ ɛ)φ 1 ] in Ω (T ɛ, ).

5 368 WEI FENG Noting that (1 δ)φ 1 (x) is the unique positive solution of the boundary value problem U + U[a 1 U δφ 1 ] = in Ω, V = on Ω, we now reach the conclusion that lim inf u(, t) (1 δ)φ 1( ) in C(Ω). We now turn our attention to the long-term survival of the the two predators and the prey in (). In the following two theorems, we will discuss the permanence effect in the predator-prey model which ensures the survival of all three species. Theorem 3. Permanence in Model () with Equal Growth Rates. Let a 1 = a = a 3 > λ and δ < 1. If and γ = l + (k 1 + l 1 )(l + δ 1) + (k 3 + l 3 )(1 + k 31 + l 31 ) < 1, (1) θ = l 33 + (k 31 + l 31 )(l 33 + δ 1) + (k 3 + l 3 )(1 + k 1 + l 1 ) < 1, (11) then the predator-prey system () is permanent. In fact, the density functions satisfy lim inf [u(t, ), v(t, ), w(t, [(1 δ)φ 1( ), (1 γ)φ 1 ( ), (1 θ)φ 1 ( in C(Ω). Proof: 1. It is seen from Theorem that when a 1 > λ, lim sup[v(, t), w(, t [(1 + k 1 + l 1 )Φ 1 ( ), (1 + k 31 + l 31 )Φ 1 ( in C(Ω). Also, for δ < 1, lim inf u(t, ) (1 δ)φ 1( ).. Hence for each < ɛ < min{1 γ, 1 θ}, there exists a T ɛ > such that and v t D v D v[a ɛ v + (k 1 + l 1 )(1 δ)φ 1 l (1 + k 1 + l 1 )Φ 1 (k 3 + l 3 )(1 + k 31 + l 31 )Φ 1 ] = D v[a 1 ɛ v γφ 1 ] in Ω (T ɛ, ) w t D 3 w D 3 w[a 3 ɛ w + (k 31 + l 31 )(1 δ)φ 1 (k 3 + l 3 )(1 + k 1 + l 1 )Φ 1 l 33 (1 + k 31 + l 31 )Φ 1 ] = D 3 w[a 1 ɛ w θφ 1 ] in Ω (T ɛ, ). 3. By the fact that (1 γ)φ 1 (x) is the unique positive solution of the boundaryvalue problem we conclude from Lemma 1 that Similarly, V + V [a 1 V γφ 1 ] = in Ω, V = on Ω, lim inf v(, t) (1 γ)φ 1( ) in C(Ω). lim inf w(, t) (1 θ)φ 1( ) in C(Ω).

6 3-SPECIES MODELS WITH DELAYS 369 Theorem. Permanence in Model () with Unequal Growth Rates. Let a 1 > a > λ, a 1 > a 3 > λ, and δ < 1. If < a 1 γ < a λ and < a 1 θ < a 3 λ, (1) with γ and θ as defined in (1) and (11), then the predator-prey system () is permanent. Proof: 1. By Theorem, when δ < 1, the u-species will persist.. Condition (1) gives γ >. for arbitrarily small ɛ >, there exists a T ɛ > such that v t D v D v[a ɛ v γφ 1 ] [a ɛ γa 1 v] in Ω (T ɛ, ). 3. Since a a 1 γ > λ, Lemma 1 and the arbitrariness of ɛ imply that lim inf v(, t) v ( ) in C(Ω), where v is the positive solution of v = v [a γa 1 v ] in Ω, v = on Ω. Therefore the v-species will also persist.. Similarly, the w-species will persist when < a 1 θ < a 3 λ. Numerical Simulations of the Model. We now work on numerical simulations [8, 5]of the three species predator-prey model () on a one-dimensional spatial domain Ω = (, 1), where the principal eigenvalue λ = π. By discretizing the differential equation systems into finite-difference systems, we obtain numerical solutions through the monotone iterative scheme developed and employed in several earlier papers[,, 6].The diffusion coefficients in () are fixed at D 1 = 1.3, D =., D 3 = 1. and the initial functions are taken to be u (x) = 1.5 sin(πx), v (x) = 1.5 sin(πx), w (x) = 5. sin(πx). We also fix each time delay term as τ ij = 1.. Example 1 - Figure 1. Persistence of u-species. As seen in Theorem, when a 1 > λ and δ < 1, the prey species u will persist. We choose the growth rates for the three species as a 1 = 13., a = 1., a 3 = 1.. The set of interaction rates l 11 =.1, l = 1., l 33 = 1., k 1 =., l 1 =., k 13 =., l 13 =.1, k 1 =.1, l 1 =.1, k 3 =.8, l 3 =.9, k 31 =.1, l 31 =.1, k 3 =.5, l 3 =.5 makes δ =.9 < 1 which ensures the persistence of the u-species. Example - Figure. Extinction of u-species. Here we give a numerical example that u-species goes to extinction for δ > 1. We let the growth rates for the three species a 1 = a = a 3 = The set of interaction rates l 11 =.3, l =.5, l 33 =.5, k 1 =., l 1 =.5, k 13 =..3, l 13 =., k 1 =., l 1 =.3, k 3 =.5, l 3 =.3, k 31 =.1, l 31 =., k 3 =., l 3 =. makes δ =.56 > 1. Example 3 - Figure 3. Permanence with equal growth rates. As stated in Theorem 3, for a 1 = a = a 3 > λ and δ, γ, θ < 1, all three species in model () will persist. We let the growth rates for the three species a 1 = a = a 3 = 11.. The set of interaction rates l 11 =.1, l =., l 33 =.3, k 1 =.1, l 1 =., k 13 =.1,

7 37 WEI FENG l 13 =.1, k 1 =., l 1 =.1, k 3 =., l 3 =.1, k 31 =.1, l 31 =., k 3 =., l 3 =. makes δ =.88, γ =.87, θ =, 87. Example - Figure. Permanence with unequal growth rates. In Theorem, we gave conditions to ensure permanence in model () with unequal growth rates: a 1 > a > λ, a 1 > a 3 > λ, δ < 1, < a 1 γ < a λ, and < a 1 θ < a 3 λ. We let the growth rates for the three species to be a 1 = 3., a =., a 3 = 19.. The set of interaction rates l 11 =.1, l =.1, l 33 =.15, k 1 =.1, l 1 =.1, k 13 =.1, l 13 =.1, k 1 =.1, l 1 =.1, k 3 =., l 3 =., k 31 =.1, l 31 =.1, k 3 =., l 3 =.1 makes δ =.58, γ =.516, θ =.56. Therefore the permanence conditions are satisfied. Fig. 1(a) U species Fig. 1(b) V species u axis v axis Fig. 1(c) W species w axis Figure 1. Persistence of u-species Fig. (a) U species Fig. (b) V species u axis 1 v axis Fig. (c) W species w axis Figure. Extinction of u-species

8 3-SPECIES MODELS WITH DELAYS 371 Fig. 3(a) U species Fig. 3(b) V species u axis v axis Fig. 3(c) W species w axis 1.5 Figure 3. Permanence with equal growth rates Fig. (a) U species Fig. (b) V species u axis 5 v axis Fig. (c) W species 3 w axis Figure. Permanence with equal growth rates 5. Discussions and Conclusions. In Theorem of Section 3, we have shown that when u-species persists in the absence of the predation from v- and w-species (a 1 > λ, l 11 < 1)and the predation interactions are not too intensive (δ < 1), the u-species at the bottom of the trophic hierarchy will persist. This persistence condition depends on all interactions rates reflecting the predation of v-species and w-species on the u-species (k 1, l 1, k 1, l 1,k 13, l 13,k 31, l 31 ). Furthermore, one can also see in Theorem 3 and Theorem that the permanence effect in the predator-prey model () requires a balanced environment on all the growth rates and interaction rates. Theorem and Theorem 3 also gives the relation between the environmental data (growth rates and interaction rates) and the ultimate lower bounds for the density functions u, v and w. by showing that lim inf u(, t) (1 δ)φ 1 ( ) (in Theorem

9 37 WEI FENG ) and similarly lim inf v(, t) (1 γ)φ 1 ( ), lim inf w(, t) (1 θ)φ 1 ( ) (in Theorem 3). Those results, together with Theorem 1, reveal further details on the effects of all those environmental data on the dynamical behavior (ultimate upper and lower bounds) for the three biological species in Model (). REFERENCES [1] W. Feng, Coexistence, stability, and limiting behavior in a one- predator-two-prey model, Journal of Mathematical Analysis and Applications, 179 (1993), [] W. Feng and X. Lu, Harmless Delays for Permanence in a class of Population Models with Diffusion Effects, Journal of Mathematical Analysis and Applications, 6 (1997), [3] W. Feng, Permanence effect in a three-species food chain model, Applicable Analysis, 5 (199), [] W. Feng and X. Lu, Some coexistence and extinction results in a three species ecological model, Differental and Integral Equations, 8 (1995), [5] X. Lu, Monotone method and convergence acceleration for finite-difference solutions of parabolic problems with time delays, Numer. Meth. Part. Diff. Eqns. 11 (1995), [6] X. Lu and W. Feng, Dynamics and numerical simulations of food chain populations, Appl. Math. Comp., 65 (199), [7] C. V. Pao, On Nonlinear Parabolic and Elliptic equations, Plenum Press, New York, 199. [8] C. V. Pao, Numerical methods for semilinear parabolic equations,, SIAM J. Numer. Anal., (1987), 35. [9] E. C. Pielou, An Introduction to Mathematical Ecology, Wiley, New York, [1] J. Wiener and K. L. Cooke, Oscillations in Systems of Differential Equations with Piecewise Constant Arguments, Journal of Mathematical Analysis and Applications, 137 (1989), Received September 6; revised May 7. address: fengw@uncw.edu