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 (SDD1 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 SDD2). 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
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.
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When the interspecific competitive strengths are small (), the opponents are relatively weak so two species can coexist (see [3, 7, 11, 12] for other models). Numerical simulation shows in our case the theoretically claimed constants , or one play as the “smallness” thresholds of the interspecific competitive strengths for coexistence. For example, in Figure 1, two species coexist if and .
Combining our analysis and numerical observation, we give some conjectures.
Conjecture. For simplicity, we ignore unstable coexistence steady states. Assume is sufficiently small. Then we conjecture the following:(1)When competes with RD as in Theorem 2 (see Table 1).(2)When RD competes with as in Theorem 3 (see Table 2).(3)When competes with as in Theorem 4 (see Table 3).
3. Proof of Theorem 1
To prove Theorem 1, we need some lemmas. In the rest of the paper, we omit the -dependence in proofs if there is no confusion.
Lemma 5 (approximation from above). Assume the hypotheses in Theorem 1. Then
Proof. Define a map by . Because is nondecreasing, is strictly increasing in so it is invertible. Now define . ThenLet . Then by the monotonicity of , the above inequality gives . Furthermore is a supersolution of (5) becauseand . Similarly, we can show that if we choose for a sufficiently small then is a subsolution of (5) and . Because the positive solution is unique, by the super- and subsolution method,By the second inequality,Also from [2, Lemma 1(i)] we knowNow suppose at some point . Then, at that point, so by (4),Combining (21) and (23), we havewhich contradicts to assumption (7). Therefore .
Lemma 6 (approximation from below). Assume the hypotheses in Theorem 1 and fix a number such that . Then the Lebesgue measure of the sethas a boundfor some positive constant independent of .
Proof. By [2, Lemma 1(ii)], we knowBy definition of and Lemma 5, the right-hand side is estimated byCombining the above inequalities, we haveNow the conclusion follows easily.
Now we are ready to prove the IFD property.
Proof of Theorem 1. (1) First we claim that has a uniform bound independent of . Because is a solution of Neumann problem (5), by the global Lipschitz regularity for Poisson’s equation (see, e.g., [13, Theorem 1.3]), we havewhere is a positive constant independent of . Because has a uniform bound independent of , so does . Furthermore, from the computationwe haveFrom this relation, we can observe that also has a uniform bound as claimed; if we divide the both sides by , then we can easily see that the gradient has a bound.
(2) Next we claim thatBy Lemmas 5 and 6, for sufficiently small , we haveHenceBy choosing or , we obtain the claim.
(3) Finally by Gagliardo-Nirenberg interpolation inequality [14, Theorem 5.9],for any . Applying the previous claims, we haveThe exponent of is maximized when and the proof is complete.
4. Proof of Theorem 2
First we introduce the diffusion pressure for the first species :DefineThen for given , and , the function can be completely determined by the equation since is a nondecreasing function. Hence, we may write , and (1) is rewritten aswhere
Let be a steady state solution of (1). The stability of is equivalent to that of the steady state of (40) for . Let and with and small. Then,and we have the following linearized eigenvalue problem:At the semitrivial steady state ,whereSo the eigenvalue value problem for (40) at is
Also consider the other semitrivial steady state and defineThen, similarly, the eigenvalue problem linearized at is
We will use the following stability criteria for semitrivial steady states. The proof is similar to that of [6, Lemma 1], but we give it here for completeness.
Lemma 7. Let be unique positive steady states of (5).(i)Let be the first eigenvalue of the eigenvalue problem If , then the semitrivial steady state of (1) is linearly unstable. If , then is linearly stable.(ii)Let be the first eigenvalue of the eigenvalue problem If , then the semitrivial steady state is linearly unstable. If , is linearly stable.
Proof. (i) First suppose . For a given , consider the eigenvalue problemand denote its first eigenvalue by . Then by (48). Moreover, will be negative when is large enough. Thus, there exists such that and the corresponding eigenfunction satisfies the first equation in (45) with . Now due to (11), it is easy to see that the operatoris invertible. Hence there is a solution of and it satisfies the second equation in (45). This yields that is an eigenvalue of the linearized problem (45), which implies that is linearly unstable.
Next, assume that . Suppose that the linearized problem (45) has a nonnegative eigenvalue with the corresponding eigenfunctions . Since is a positive solution of (11), the operator has only strictly negative eigenvalues. But if , by the second equation in (45), the operator has a nonnegative eigenvalue , which is a contradiction. Hence . However, by the Rayleigh quotient of (48) and (45), this yields thatThis is a contradiction. Therefore, is linearly stable.
(ii) Consider the linearized eigenvalue problem of (5) at , which is written asSince is a unique positive steady state, it is linearly neutrally stable (in fact it is globally asymptotically stable [2, Theorem 2]). Hence every eigenvalue of the linear operator is nonpositive. Thus, if , because of , the operatorhas strictly negative eigenvalues and thus it is invertible. Let be an eigenpair of (49) and be the solution ofThen, is an eigenpair of the linearized problem (47), which implies that is linearly unstable.
Since an eigenvalue of (47) is also an eigenvalue of (49), eigenvalues of (47) are all strictly negative if and hence is linearly stable.
Now we are ready to prove Theorem 2. Due to Lemma 7, it is sufficient to compute the sign of the first eigenvalues and .
Proof of Theorem 2. (i) RecallWhen , by assumption (7), we can show that (see the proof of [6, Theorem 1]). On the other hand, because is a nondecreasing function, and decrease strictly as increases. By the continuity of , there exists a constant such that is linearly unstable for all and is linearly stable for all .
Furthermore, since (one may check this using the maximum principle), when , in . Thus, for all . This implies that .
(ii) RecallBy Theorem 1, as . Therefore, for a given , there is a such thatHence for any ,which implies .
Similarly, for a given , there is a such thatThen with ,(iii) Suppose there is a positive coexistence steady state . Then by (40),Considering the Rayleigh quotient of this equation, we haveSimilarly we also haveChoosing in (63),Now observe the sign of the integrand. If , so the integrand vanishes. Hence the integral is actually taken over the set where , and then the last integral is positive. This is a contradiction so there is no coexistence steady state.
5. Proof of Theorem 3
In this section, we consider the following system:where . We introduce the diffusion pressure for the second species :where is a unique positive solution of (5). Definethenand (66) is rewritten aswhere . The corresponding eigenvalue value problems linearized at the semitrivial steady states and areandrespectively.
Omitting the proof, we use the following stability criteria for semitrivial steady states; the proof is similar to the one of Lemma 7.
Lemma 8. Let be unique positive steady states of (5).(i)Let be the first eigenvalue of If , then the semitrivial steady state of (66) is linearly unstable. If , then is linearly stable.(ii)Let be the first eigenvalue of If , then is linearly unstable. If , is linearly stable.
Because system (66) has no cross-diffusion (a cross-diffusion is a Fokker-Planck diffusion depending on both and ), we can prove that it is a strongly monotone dynamical system.
Lemma 9 (monotonicity). Let and be solutions of (66) with initial data and , respectively. If and , then and .
We omit the proof; it is a slight modification of the proof in [8, pp.266–267].
Now we give a proof Theorem 3. Due to Lemma 8, it suffices to check the sign of and for the linear stability.
Proof of Theorem 3. (i) Since decreases strictly as increases. If , then and .
If , choosing in the above Rayleigh quotient, we obtainTherefore, there exists a constant such that , when , and when .
(ii) The proof is the same as the one for Theorem 2 (ii).
(iii) By (i), (ii), and the assumption, the semitrivial steady states and are either both stable or both unstable. Hence there is a coexistence steady state by virtue of the monotonicity of system [15].
(iv) By an argument almost similar to the proof of Theorem 2 (iii), we can show there is no coexistence steady state. Also is linearly stable by (ii). Therefore by the theory of monotone dynamical system [16, Proposition 9.1 and Theorem 9.2], the linearly stable semitrivial steady state is globally asymptotically stable as in [15, 17].
6. Proof of Theorem 4
We consider the following system:where and . SetThen the corresponding eigenvalue problems linearized at the semitrivial steady states and areandrespectively.
In a similar way to the proofs of Lemmas 7 and 8, we obtain the following stability criterion. We omit the proof.
Lemma 10. Let be unique positive steady states of (5).(i)Let be the first eigenvalue of If , then the semitrivial steady state of (77) is linearly unstable. If , is linearly stable.(ii)Let be the first eigenvalue of If , then is linearly unstable. If , is linearly stable.
Proof of Theorem 4. (i), (ii) The proof is similar to the proofs of Theorems 2 and 3.
(iii) If there is a coexistence steady state ,Choosing ,If we showis strictly positive, it causes a contradiction so the proof is complete.
Suppose . Then soSimilarly, if and , thenThus .
Finally suppose and . Note thatSo if and , we haveWhen , by the structure of in (4), . HenceThis completes the proof of Theorem 4.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
J. Chung was supported by National Institute for Mathematical Sciences (NIMS) grant funded by Korean Government (A23100000) and KERI Primary research program through the National Research Council of Science & Technology (NST) funded by the Ministry of Science and ICT (MSIT) (no. 18-12-N0101-34). O. Kwon was supported by Young Researcher Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. NRF-2016R1C1B2014942). This work was supported by the intramural research grant of Chungbuk National University in 2015.