Invasion traveling wave solutions of a competitive system with dispersal
© Pan and Lin; licensee Springer 2012
Received: 31 January 2012
Accepted: 8 October 2012
Published: 24 October 2012
This paper is concerned with the invasion traveling wave solutions of a Lotka-Volterra type competition system with nonlocal dispersal, the purpose of which is to formulate the dynamics between the resident and the invader. By constructing upper and lower solutions and passing to a limit function, the existence of traveling wave solutions is obtained if the wave speed is not less than a threshold. When the wave speed is smaller than the threshold, the nonexistence of invasion traveling wave solutions is proved by the theory of asymptotic spreading.
MSC: 35C07, 35K57, 37C65.
Keywordscomparison principle asymptotic spreading upper and lower solutions invasion waves
in which all the parameters are positive and , , , are two competitors. Many investigators considered its traveling wave solutions connecting different spatial homogeneous steady states such as the existence, monotonicity, minimal wave speed and stability; see [1–16].
In particular, if holds in (1.1), then the corresponding reaction system has a stable equilibrium and an unstable one . With the condition , many papers including [2, 3, 5, 6, 8, 16] studied the traveling wave solutions connecting with . These traveling wave solutions can formulate the spatial exclusive process between the resident and the invader so that the minimal wave speed reflecting the invasion speed of the invader becomes a hot topic in these works; we refer to Shigesada and Kawasaki  for some examples of the corresponding biological records and the literature importance of invasion speed. Moreover, the similar problem was also discussed in different spatial media such as the lattice differential systems in Guo and Liang , Guo and Wu .
, , are probability functions formulating the random dispersal of individuals and satisfy the following assumptions:
(J1) is nonnegative and Lebesgue measurable for each ;
(J2) for any , , ;
(J3) , , , .
In (1.2), the spatial migration of individuals is formulated by the so-called dispersal operator, which has significant sense in population dynamics. For example, in the patch models of population dynamics , the rate of immigration into a patch from a particular other patch is usually taken as proportional to the local population, and the dispersal can be regarded as the extension of these ideas to a continuous media model. Such a diffusion mechanism also arises from physics processes with long range effect and other disciplines , and the dynamics of evolutionary systems with dispersal effect has been widely studied in recent years; we refer to [13, 21–32] and the references cited therein.
From the viewpoint of ecology, a traveling wave solution satisfying (1.4)-(1.5) can model the population invasion process: at any fixed , only (the resident) can be found long time ago ( such that ), but after a long time ( such that ), only (the invader) can be seen. Therefore, we call a traveling wave solution satisfying (1.4)-(1.5) an invasion traveling wave solution.
The rest of this paper is organized as follows. In Section 2, we give some preliminaries. By constructing upper and lower solutions and using a limit process, the existence of traveling wave solutions is established in Section 3. In the last section, we obtain the nonexistence of traveling wave solutions.
Clearly, a fixed point of in X satisfies (2.1), and a solution of (2.1) is also a fixed point of F. To continue our discussion, we also introduce the following definition.
for , then it is an upper (a lower) solution of (2.1).
Using Pan et al. , Theorem 3.2, we obtain the following conclusion.
Lemma 2.2 Assume that is an upper solution of (2.1), while is a lower solution of (2.1). Also, suppose that
(P2) , ;
(P3) for all , , and .
In Jin and Zhao , the authors investigated the asymptotic spreading of a periodic population model with spatial dispersal. Note that the parameters in (2.4) are positive constants, then , Theorem 2.1, implies the following result.
Then Jin and Zhao , Theorem 3.5, indicates the following conclusion.
where is defined by (2.4).
3 Existence of traveling wave solutions
for any , .
For each , has two positive real roots .
If , then there exists such that and for any .
If , then for any .
The above result is clear and we omit the proof here. Using these constants, we can prove the following conclusion.
Then (2.1)-(2.2) has a monotone solution.
Claim A: is an upper solution to (2.1).
Evidently, is a lower solution to (2.1) (for the existence of and , we refer to Pan et al. ). By Lemma 2.2, we see that (2.1)-(2.2) has a monotone solution . Now, it suffices to prove Claim A.
which completes the proof on for .
Therefore, Claim A is true. The proof is complete. □
Then (2.1)-(2.2) has a monotone solution with .
and the convergence in s is uniform for . Letting and using the dominated convergence theorem in , we know that also satisfies (2.1) with . In addition, the following items are also clear.
(T1) (by (3.5));
(T2) , are nondecreasing in ξ;
(T3) , , .
Using the dominated convergence theorem in for , we get the following possible conclusions:
which is also a contradiction. What we have done implies that . Using the dominated convergence theorem in again, we see that and .
has a monotone solution, which is impossible. Therefore, holds.
Thus, is a positive monotone solution of (2.1)-(2.2) with , the proof is complete. □
4 Nonexistence of traveling wave solutions
In this section, we shall formulate the nonexistence of invasion traveling wave solutions of (1.2) by the theory of asymptotic spreading. Before this, we first present a comparison principle formulated by Jin and Zhao , Theorem 2.3.
then , , .
We now give the main result of this section.
Theorem 4.2 If , then (2.1)-(2.2) has no positive solutions.
Then is evident.
which implies a contradiction between (4.6) and (4.7). The proof is complete. □
Remark 4.3 Under proper assumptions, we have obtained the threshold of the existence of positive solutions to (2.1)-(2.2).
The authors express their thanks to the referees for their helpful comments and suggestions on the manuscript. This work was partially supported by the Development Program for Outstanding Young Teachers in Lanzhou University of Technology (1010ZCX019), NSF of China (11101094) and FRFCU (lzujbky-2011-k27).
- Ahmad S, Lazer AC, Tineo A: Traveling waves for a system of equations. Nonlinear Anal. TMA 2008, 68: 3909-3912. 10.1016/j.na.2007.04.029MathSciNetView ArticleGoogle Scholar
- Fei N, Carr J: Existence of travelling waves with their minimal speed for a diffusing Lotka-Volterra system. Nonlinear Anal., Real World Appl. 2003, 4: 503-524. 10.1016/S1468-1218(02)00077-9MathSciNetView ArticleGoogle Scholar
- Gourley SA, Ruan S: Convergence and traveling fronts in functional differential equations with nonlocal terms: a competition model. SIAM J. Math. Anal. 2003, 35: 806-822. 10.1137/S003614100139991MathSciNetView ArticleGoogle Scholar
- Guo JS, Liang X: The minimal speed of traveling fronts for the Lotka-Volterra competition system. J. Dyn. Differ. Equ. 2011, 23: 353-363. 10.1007/s10884-011-9214-5MathSciNetView ArticleGoogle Scholar
- Hosono Y: The minimal speed of traveling fronts for a diffusive Lotka-Volterra competition model. Bull. Math. Biol. 1998, 60: 435-448. 10.1006/bulm.1997.0008View ArticleGoogle Scholar
- Huang W: Problem on minimum wave speed for a Lotka-Volterra reaction-diffusion competition model. J. Dyn. Differ. Equ. 2010, 22: 285-297. 10.1007/s10884-010-9159-0View ArticleGoogle Scholar
- Kan-on Y, Fang Q: Stability of monotone travelling waves for competition-diffusion equations. Jpn. J. Ind. Appl. Math. 1996, 13: 343-349. 10.1007/BF03167252MathSciNetView ArticleGoogle Scholar
- Lewis MA, Li B, Weinberger HF: Spreading speed and linear determinacy for two-species competition models. J. Math. Biol. 2002, 45: 219-233. 10.1007/s002850200144MathSciNetView ArticleGoogle Scholar
- Li WT, Lin G, Ruan S: Existence of traveling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems. Nonlinearity 2006, 19: 1253-1273. 10.1088/0951-7715/19/6/003MathSciNetView ArticleGoogle Scholar
- Lin G, Li WT: Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays. J. Differ. Equ. 2008, 244: 487-513. 10.1016/j.jde.2007.10.019View ArticleGoogle Scholar
- Lin G, Li WT, Ma M: Travelling wave solutions in delayed reaction diffusion systems with applications to multi-species models. Discrete Contin. Dyn. Syst., Ser. B 2010, 19: 393-414.MathSciNetGoogle Scholar
- Lv G, Wang M: Traveling wave front in diffusive and competitive Lotka-Volterra system with delays. Nonlinear Anal., Real World Appl. 2010, 11: 1323-1329. 10.1016/j.nonrwa.2009.02.020MathSciNetView ArticleGoogle Scholar
- Murray LD: Mathematical Biology. Heidelberg, Springer; 1989.View ArticleGoogle Scholar
- Tang MM, Fife P: Propagating fronts for competing species equations with diffusion. Arch. Ration. Mech. Anal. 1980, 73: 69-77. 10.1007/BF00283257MathSciNetView ArticleGoogle Scholar
- Wang M, Lv G: Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delays. Nonlinearity 2010, 23: 1609-1630. 10.1088/0951-7715/23/7/005MathSciNetView ArticleGoogle Scholar
- Yuan Z, Zou X: Co-invasion waves in a reaction diffusion model for competing pioneer and climax species. Nonlinear Anal., Real World Appl. 2010, 11: 232-245. 10.1016/j.nonrwa.2008.11.003MathSciNetView ArticleGoogle Scholar
- Shigesada N, Kawasaki K: Biological Invasions: Theory and Practice. Oxford University Press, Oxford; 1997.Google Scholar
- Guo JS, Wu CH: Traveling wave front for a two-component lattice dynamical system arising in competition models. J. Differ. Equ. 2012, 252: 4367-4391.MathSciNetGoogle Scholar
- Yu Z, Yuan R: Travelling wave solutions in nonlocal reaction-diffusion systems with delays and applications. ANZIAM J. 2009, 51: 49-66. 10.1017/S1446181109000406MathSciNetView ArticleGoogle Scholar
- van den Driessche P: Spatial structure: patch models. In Mathematical Epidemiology. Edited by: Brauer F, Driessche P, Wu J. Springer, Berlin; 2008:179-189.View ArticleGoogle Scholar
- Bates PW: On some nonlocal evolution equations arising in materials science. Fields Inst. Commun. 48. In Nonlinear Dynamics and Evolution Equations. Edited by: Brunner H, Zhao X, Zou X. Amer. Math. Soc., Providence; 2006:13-52.Google Scholar
- Bates PW, Fife PC, Ren X, Wang X: Traveling waves in a convolution model for phase transition. Arch. Ration. Mech. Anal. 1997, 138: 105-136. 10.1007/s002050050037MathSciNetView ArticleGoogle Scholar
- Bates PW, Han J: The Neumann boundary problem for a nonlocal Cahn-Hilliard equation. J. Differ. Equ. 2005, 212: 235-277. 10.1016/j.jde.2004.07.003MathSciNetView ArticleGoogle Scholar
- Bates PW, Zhao G: Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal. J. Math. Anal. Appl. 2007, 332: 428-440. 10.1016/j.jmaa.2006.09.007MathSciNetView ArticleGoogle Scholar
- Carr J, Chmaj A: Uniqueness of travelling waves for nonlocal monostable equations. Proc. Am. Math. Soc. 2004, 132: 2433-2439. 10.1090/S0002-9939-04-07432-5MathSciNetView ArticleGoogle Scholar
- Cortázar C, Coville J, Elgueta M, Martínez S: A nonlocal inhomogeneous dispersal process. J. Differ. Equ. 2007, 241: 332-358. 10.1016/j.jde.2007.06.002View ArticleGoogle Scholar
- Ermentrout B, Mcleod J: Existence and uniqueness of travelling waves for a neural network. Proc. R. Soc. Edinb. A 1994, 123: 461-478.MathSciNetView ArticleGoogle Scholar
- Fife PC: Some nonclassical trends in parabolic and parabolic-like evolutions. In Trends in Nonlinear Analysis. Edited by: Kirkilionis M, Krömker S, Rannacher R, Tomi F. Springer, Berlin; 2003:153-191.View ArticleGoogle Scholar
- Kao CY, Lou Y, Shen W: Random dispersal vs. non-local dispersal. Discrete Contin. Dyn. Syst. 2010, 26: 551-596.MathSciNetGoogle Scholar
- Shen W, Zhang A: Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats. J. Differ. Equ. 2010, 249: 747-795. 10.1016/j.jde.2010.04.012MathSciNetView ArticleGoogle Scholar
- Yanagida E, Zhang L: Speeds of traveling waves in some integro-differential equations arising from neuronal networks. Jpn. J. Ind. Appl. Math. 2010, 27: 347-373. 10.1007/s13160-010-0021-xMathSciNetView ArticleGoogle Scholar
- Zhang G: Global stability of wavefronts with minimal speeds for nonlocal dispersal equations with degenerate nonlinearity. Nonlinear Anal. 2011, 74: 6518-6529. 10.1016/j.na.2011.06.035MathSciNetView ArticleGoogle Scholar
- Pan S, Li WT, Lin G: Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications. Z. Angew. Math. Phys. 2009, 60: 377-392. 10.1007/s00033-007-7005-yMathSciNetView ArticleGoogle Scholar
- Jin Y, Zhao XQ: Spatial dynamics of a periodic population model with dispersal. Nonlinearity 2009, 22: 1167-1189. 10.1088/0951-7715/22/5/011MathSciNetView ArticleGoogle Scholar
- Coville J, Dupaigne L: Propagation speed of travelling fronts in nonlocal reaction-diffusion equation. Nonlinear Anal. TMA 2005, 60: 797-819. 10.1016/j.na.2003.10.030MathSciNetView ArticleGoogle Scholar
- Coville J, Dupaigne L: On a non-local equation arising in population dynamics. Proc. R. Soc. Edinb. A 2007, 137: 725-755.MathSciNetView ArticleGoogle Scholar
- Li WT, Sun Y, Wang ZC: Entire solutions in the Fisher-KPP equation with nonlocal dispersal. Nonlinear Anal., Real World Appl. 2010, 11: 2302-2313. 10.1016/j.nonrwa.2009.07.005MathSciNetView ArticleGoogle Scholar
- Lv G: Asymptotic behavior of traveling fronts and entire solutions for a nonlocal monostable equation. Nonlinear Anal. TMA 2010, 72: 3659-3668. 10.1016/j.na.2009.12.047View ArticleGoogle Scholar
- Pan S: Traveling wave fronts of delayed non-local diffusion systems without quasimonotonicity. J. Math. Anal. Appl. 2008, 346: 415-424. 10.1016/j.jmaa.2008.05.057MathSciNetView ArticleGoogle Scholar
- Pan S, Li WT, Lin G: Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay. Nonlinear Anal. TMA 2010, 72: 3150-3158. 10.1016/j.na.2009.12.008MathSciNetView ArticleGoogle Scholar
- Sun Y, Li WT, Wang ZC: Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity. Nonlinear Anal. TMA 2011, 74: 814-826. 10.1016/j.na.2010.09.032MathSciNetView ArticleGoogle Scholar
- Wu S, Liu S: Traveling waves for delayed non-local diffusion equations with crossing-monostability. Appl. Math. Comput. 2010, 217: 1435-1444. 10.1016/j.amc.2009.05.056MathSciNetView ArticleGoogle Scholar
- Xu Z, Weng P: Traveling waves in a convolution model with infinite distributed delay and non-monotonicity. Nonlinear Anal., Real World Appl. 2011, 12: 633-647. 10.1016/j.nonrwa.2010.07.006MathSciNetView ArticleGoogle Scholar
- Zhang G, Li WT, Wang ZC: Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity. J. Differ. Equ. 2012, 252: 5096-5124. 10.1016/j.jde.2012.01.014MathSciNetView ArticleGoogle Scholar
- Li X, Lin G: Traveling wavefronts in nonlocal dispersal and cooperative Lotka-Volterra system with delays. Appl. Math. Comput. 2008, 204: 738-744. 10.1016/j.amc.2008.07.016MathSciNetView ArticleGoogle Scholar
- Zhang G, Li WT, Lin G: Traveling waves in delayed predator-prey systems with nonlocal diffusion and stage structure. Math. Comput. Model. 2009, 49: 1021-1029. 10.1016/j.mcm.2008.09.007MathSciNetView ArticleGoogle Scholar
- Thieme HR, Zhao XQ: Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction diffusion models. J. Differ. Equ. 2003, 195: 430-470. 10.1016/S0022-0396(03)00175-XMathSciNetView ArticleGoogle Scholar