Journal of Function Spaces

Volume 2018, Article ID 7879598, 12 pages

https://doi.org/10.1155/2018/7879598

## Dynamics of Lotka-Volterra Competition Systems with Fokker-Planck Diffusion

^{1}Thermoelectric Conversion Research Center, Korea Electrotechnology Research Institute, 12 Bulmosan-ro 10beon-gil, Seongsan-gu, Changwon-si, Gyeongsangnam-do 51543, Republic of Korea^{2}Department of Mathematics, Chungbuk National University, Chungdae-ro 1, Seowon-Gu, Cheongju, Chungbuk 28644, Republic of Korea

Correspondence should be addressed to Ohsang Kwon; rk.ca.kubgnuhc@nowkgnasho

Received 25 May 2018; Accepted 5 August 2018; Published 2 September 2018

Academic Editor: Dumitru Motreanu

Copyright © 2018 Jaywan Chung and Ohsang Kwon. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider competition systems of two species which have different dispersal strategies and interspecific competitive strengths. One of the dispersal strategies is random dispersal and the other is a Fokker-Planck diffusion whose motility is piecewise constant and jumps up when the resource is not enough. In this paper, first we show the Fokker-Planck diffusion allows ideal free distribution. Next we show the linear stability of semitrivial steady states is determined exactly by a threshold on the interspecific competitive strengths. Some conditions for coexistence and global asymptotic stability are also provided.

#### 1. Introduction and Main Result

In this paper, we study Lotka-Volterra competition systems with Fokker-Planck diffusion of the following form:where is a smooth bounded domain in and a given resource distribution is a nonconstant -function satisfying . The motility functions of Fokker-Planck diffusion are assumed to depend only onrespectively. High values of mean a scarcity of resources; the difference between them is that depends on the competitive strength of other species while does not. The nonnegative constants are interspecific competitive strengths. The intraspecific competitive strengths are assumed to be one.

We consider two types of motility functions. The first one is the random dispersal (RD) where ’s are constant. The other is based on the following step function: where and are positive constants satisfying . For a single species case, if we let , the motility function models the situation where organisms drastically change their departing probability in random walk from to exactly when the resource becomes insufficient. This motility function is called* starvation-driven diffusion* (SDD) [1]. Unfortunately, due to the jump discontinuity, the Laplacian of is not well-defined. So we consider a regularized function which is a* nondecreasing* smooth function such that andfor given . For example, we may choose where is a standard mollifier supported in .

The second motility function is given by this . In such cases we may choose or . For simplicity we will call the first one and the second one .

Our first main result shows why the SDD is worthy of notice; it is an efficient dispersal strategy helping the steady state be the* ideal free distribution* (IFD).

Theorem 1 (approximate IFD). *Suppose there is a unique positive solution ofwhereIf the resource distribution satisfiesthen*

The uniqueness assumption in Theorem 1 can be replaced by some sufficient conditions. For example, a condition on the resource distribution,guarantees the uniqueness [2, Theorem 3(i)]. There are other sufficient conditions restricting the shape of [2, Theorem 2] or the shape of [2, Theorem 3(ii)].

Condition (7) seems not to be a real restriction because it is necessary even when there is no logistic growth term. In [2, Theorem 5], with no logistic growth term, it was shown thatunder assumption (7) although their choice of is explicit and slightly different from ours; for example, they assume but we assume . Furthermore it was suggested that condition (7) seems almost optimal [2, Remark 3].

With the logistic term the same IFD property (10) is proved in [2, Theorem 6]. But it needs a restrictive condition on [2, equation ] and relies on the explicit form of . Although our decay order in (8) is worse than the order in (10), our theorem does not have such restrictions.

In the sense of Theorem 1, the motility function gives a best possible dispersal strategy in single species case; the steady state population distribution is very close to the resource distribution and furthermore the steady state is globally asymptotically stable [2, Theorem 2].

Next we consider competitions between two dispersal strategies among RD, , and . For instance, to observe the competition between and RD, we let and for some positive constant in (1). Then system (1) can have several steady states such as the trivial steady state and semitrivial steady states and . Here is a positive solution of (5) and is a unique positive solution ofInspecting the linear stability of each steady state, we can understand which dispersal strategy is more advantageous.

In the competition between and RD, the linear stability of the semitrivial steady state , which means RD prevails and goes extinct, changes exactly across a threshold on RD’s interspecific competitive strength. Such a threshold exists for the linear stability of but it is one so less than . This suggests that gains an advantage over RD since the RD needs more interspecific competitive strength to prevail. In the competition between RDs, the RD could prevail with smaller interspecific competitive strength [3, Theorem 1.1].

Theorem 2 (SDD_{1} versus RD). *Assume the hypotheses in Theorem 1 and let with and . Then we obtain the following:*(i)*Assume is sufficiently small. There exists a constant such that the semitrivial steady state is linearly stable if and is linearly unstable if . Furthermore,*(ii)*Assume is sufficiently small. If , then is linearly stable. If , then is linearly unstable.*(iii)*When , , and , there is no positive coexistence steady state for any .*

*In the competition between RDs, the one with smaller diffusivity always prevails [4, 5] if . Also even if is slightly less than one ( has weak strength), a lower diffusivity of ( has strong dispersal strategy) can overcome the disadvantage and make prevail [3].*

*But in the competition between and RD, Theorem 2 (i) shows that RD cannot prevail no matter how small diffusivity it has if . When , the same observation was given in [6, Theorem 1]. Furthermore Theorem 2 (ii) claims that loses its advantage as soon as is smaller than one. This means that low interspecific competitive strength of cannot be overcome by the dispersal strategy, contrary to the RD case. This is unexpected since the seems to be more competitive than RD; the can build up the ideal free distribution while the RD cannot.*

*In the competition between RD and , the dynamics is similar to the case of Theorem 2 but is less competitive than .*

*Theorem 3 (RD versus SDD _{2}). Assume the hypotheses in Theorem 1 and let and with . Then we obtain the following:(i)Assume is sufficiently small. There exists a constant such that the semitrivial steady state is linearly stable if and is linearly unstable if . Furthermore,(ii)Assume is sufficiently small. If , then is linearly stable. If , then is linearly unstable.(iii)With the in (i), suppose either and or and . Then there is a coexistence steady state for all sufficiently small .(iv)If and , then is globally asymptotically stable for all sufficiently small .*

*Since for any [7, Theorem 1.2], is not necessarily bigger than one. So smaller interspecific competitive strength is enough for RD to prevail against than the strength required against is.*

*By Theorem 3 (iv), we can see that when . When , it is known that if , is globally asymptotically stable; if , is linearly stable [8, Theorem 2.1].*

*Finally, we consider the competition between and .*

*Theorem 4 (SDD with versus SDD with ). Assume the hypotheses in Theorem 1 and let and where ; . Then for all sufficiently small , we obtain the following:(i)If , then is linearly stable. If , then is linearly unstable.(ii)If , then is linearly stable. If , then is linearly unstable.(iii)If , then there is no coexistence steady state.*

*Although we expect to be more competitive than , our linear stability analysis does not indicate it. We solve system (1) numerically to observe it in the next section (see Figure 3).*

*The proofs of the theorems are given in the subsequent sections.*

*2. Numerical Simulation and Conjecture*

*2. Numerical Simulation and Conjecture**In this section, by numerically solving the competition system (1), we observe how competitive the dispersal strategies RD, , and are.*

*We fix the following parameters:To construct the motility function , we choose the following mollifier whose support is : where is the normalization constant which makes . Let and . Then our motility function is given byand .*

*To numerically solve the competition system (1), we use the pdepe function of MATLAB [9] which implements a method of lines [10] with 100 evenly spaced spatial grid points. The mass ratio of , , at is plotted in Figures 1, 2, and 3 for the situations of Theorems 2, 3, and 4, respectively.*