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Research Article | Open Access
Minimal Wave Speed of Bacterial Colony Model with Saturated Functional Response
By considering bacterium death and general functional response we develop previous model of bacterial colony which focused on the traveling speed of bacteria. The minimal wave speed for our model is expressed by parameters and the necessary and sufficient conditions for traveling wave solutions (TWSs) are given. To prove the existence of TWSs, an auxiliary system is introduced and the existence of TWSs for this auxiliary system is proved by Schauder’s fixed point theorem. The limit arguments show the existence of TWSs for original system. By introducing negative one-sided Laplace transform, we prove the nonexistence of TWSs.
Experiments show that bacterial colonies on agar plates with nutrients exhibit a variety of sizes and shapes [1–7]. According to the substrate softness and nutrient concentration, the colony patterns are divided into five types [6, 8]. Why were so many rich diffusive patterns observed in bacterial experiments? To answer this question, lots of diffusive mathematical models have been proposed and studied [4, 7, 9–16]. In these mathematical models, the colony patterns are proved or simulated on bounded domains. For bacterial colony, the colony speed is one of the most important focuses and traveling wave solution (TWS) can foresee such speed. Thus many researches studied the bacterial colony speeds through TWSs [17–24].
To more exactly anticipate the traveling speed of bacterial colony, we develop above TWS models to a more accurate bacterial colony model with bacterium death and general functional response, which is more complex compared with above TWS models. Let and denote the concentrations of nutrients and bacteria at time and position , respectively. Then our model is as follows: where parameters and denote the motility of the nutrients and bacteria. is the conversion rate of nutrients to bacteria and is the death rate of bacteria. Function is the functional response to nutrients. For simplicity, we assume with and . Actually, in the following proof we only use the monotonicity and boundedness of .
In this paper, the minimal wave speed is given and the necessary and sufficient conditions for the existence of TWSs are obtained. To arrive at such aim, the existence of TWSs is proved by Schauder’s fixed point theorem and the nonexistence is finished by negative one-sided Laplace transform proposed firstly by us. To apply Schauder’s fixed point theorem, a bounded invariant cone is needed. Such cone is constructed generally by a pair of upper and lower solutions. However, it is difficult for us to construct such solutions for model (1). Consequently, an auxiliary system is introduced, for which the upper and lower solutions can be easily constructed and are very simple. Such type of upper and lower solutions is motivated by Diekmann . Then limit arguments give the existence of TWSs of model (1). Two-sided Laplace transform was firstly introduced by Carr and Chmaj  to prove nonexistence of TWSs and was further applied by [27–29]. However, the introduction of negative one-sided Laplace transform simplifies the proof.
This paper is organized as follows. In the next section, an auxiliary system is firstly introduced and the existence of TWSs is proved by Schauder’s fixed point theorem. Then limit arguments give the existence of TWSs for original system. In Section 3, the negative one-sided Laplace transform is defined and then the nonexistence of TWSs is obtained.
2. Existence of Traveling Wave Solution
A traveling wave solution of system (1) is a nonnegative nontrivial solution of the form satisfying boundary condition where is initial density of nutrients. It is obvious that .
Define . The existence of traveling wave solutions is given as follows.
Substituting wave profile into system (1) yields the following equations: where denotes the derivative with respect to .
To prove the existence of solutions of (5) satisfying (3), we construct an auxiliary system: where is a positive constant and can be supposed to be small enough according to what we will need. Next, an invariant cone will be constructed and Schauder’s fixed point theorem will be used to prove the existence of traveling wave solutions. We firstly linearize the second equation of (6) at and obtain Obviously, the characteristic equation is Denote and . In the remainder of this section, we always suppose and hold unless other conditions are specified. Define where and .
Lemma 2. The function satisfies inequality for any .
Proof. Firstly, assume and, therefore, . Since satisfies (7), we have Secondly, let , which implies . We have that The proof is completed.
Lemma 3. For and , the function satisfies for any .
Proof. It is easy to show that . When , then and the lemma is obviously true. Now, suppose . Then and Thus the proof is completed.
Lemma 4. Let . Then for large enough, the function satisfies for any .
Proof. It is clear that if and only if , that if and only if , and that if and only if . Assume . When , then , , and Lemma 4 holds.
In this paragraph, assume . Then , , and . To prove this lemma, it is enough to show where . Since , we only need to show Since by and for any , we have Since , inequality (17) is satisfied if The proof is completed.
To apply Schauder’s fixed point theorem, we will introduce a topology in . Let be the roots of and the roots of where and are positive constants that will be determined later. Let be a positive constant which can be small enough. For , define We will find the traveling wave solution in the following profile set: Obviously, is closed and convex in . Firstly, we change system (6) into the following form: where , , and Furthermore, define by where .
Lemma 5. Consider .
Proof. Suppose ; that is, for any . Then we will prove that
for any .
If , then , which implies that since . Assume . From Lemma 4 and , it is clear that which implies that where the final inequality is due to and . In conclusion, for any .
Similarly, it can be proved that for any . The proof is completed.
Lemma 6. For small enough, map is continuous with respect to the norm in .
Proof. Suppose , which implies for any , where . Then we have where is between and and Therefore, Set . If , it holds that If , we have Consequently, we conclude that where Thus is continuous with respect to the norm in . Similarly, it can be proved that is also continuous with respect to the norm in . The proof is completed.
Lemma 7. Map is compact with respect to the norm in .
Proof. Assume . Then we have
Consequently, is bounded. Similarly, is also bounded, which shows that is uniformly bounded and equicontinuous with respect to the norm .
Furthermore, for any positive integer , we define Obviously, for fixed , is uniformly bounded and equicontinuous with respect to the norm in , implying that is a compact operator. Since we have Similarly, we can prove that when . Thus, when . By Proposition 2.1 in Zeidler  we see that converges to in with respect to the norm . Consequently, is compact with respect to the norm . The proof is completed.
Proof. Combination of Schauder’s fixed point theorem, Lemmas 5, 6, and 7 shows that there exists a nonnegative traveling wave solution such that when . Since is the fixed point of , L’Hospital principal shows that . Then from (6) we have that . Since is the solution of (6), thus
The first equation of (48) can be changed into
Multiplying this equation by yields
From the proof of Lemma 7, we have is bounded in . Then integrating above equality from to , we have
which implies that is nonincreasing in and has limit as . By the definition of and there is a such that and when . Therefore, if , we have that which implies that .
Integrating the first equation of (48) from to gives which implies that . Integrating the second equation of (48) from to gives Thus and since is bounded in . By (51) and L’Hospital principal, it follows . Then using (52) and (53) shows that
Next, we prove that . Let It is clear that and . Define . Calculations show that Multiplying this equality by and then integrating from to show that for any . Consequently, is nondecreasing in . Since we have that for any . The proof is completed.
Proof of Theorem 1. Firstly, we consider the case . Let be a sequence such that and . By Lemma 8, there exists a traveling wave solution of system (6) for satisfying the conclusion of Theorem 1. From (51), we have
Similarly, it can be shown that , where is independent of due to . By (6), there is a positive constant independent of such that , , , and are bounded in by .
Therefore, , , are equicontinuous and uniformly bounded in . Then Arzela-Ascoli’s theorem implies that there exists a subsequence such that uniformly in any bounded closed interval when and pointwise on , where . Since is the solution of (6) and , we get That is, is a solution of (5) satisfying (3):
To complete the proof of case , we need to prove . Integrating the second equation of system (5) from to and noting that is decreasing from to , we have which implies .
To prove case , let the parameter in system (5), , and . Similar to above proof about case , we can finish the proof.
3. Nonexistence of Traveling Wave Solution
In this section, we give the conditions on which system (1) has no traveling wave solutions.
Theorem 9. (I) Assume . Then for any , system (1) has no nonnegative traveling wave solutions satisfying boundary condition (3).
(II) Suppose . Then for any , system (1) has no traveling wave solutions satisfying boundary condition (3).
Proof of Theorem 9(I). Suppose (I) fails. That is, system (5) has a nonnegative nontrivial traveling wave solution satisfying boundary condition (3). Since and , there exists a such that for any . Thus, we get
for any . That is,
for any . Now we show . Denote . From the second equation of (5), we have
where . Since satisfies boundary condition (3), it follows . If , then or , which imply or contradicting .
Defining and integrating (65) from to , we have that Integrating (67) from to with yields Therefore, we get for any . Since is increasing in , it is clear that for any and . Therefore, there is large enough such that for any . Let and . We get that for any . Since is bounded in , thus as , which implies that there exists such that for any . Hence, we have that for and that there exists such that . In addition, inequalities (65)–(68) imply that
To complete the proof, we define negative one-sided Laplace transform as follows: for . Obviously is increasing in such that satisfying or . Since , we have . Trivial calculations show that satisfies The second equation of (5) can be rewritten as where Define . Noticing yields . Since (5) is autonomous, then for any , is also a solution of (5) satisfying boundary condition (3) and as . Hence, without losing generality we can assume for all . That is, Applying the operator to this inequality and using the properties of concluded above yield that where is the characteristic function of (7) and Consequently, we have If , then and, therefore, , which is a contradiction. If , we have that by the monotonicity of and the definitions of and , which is still a contradiction. The proof of Theorem 9(I) is completed.
Proof of Theorem 9(II). Suppose is a nontrivial solution of system (5) satisfying boundary condition (3). Similar to the arguments about (66), it is easy to show that . Then integrating the second equation of (5) from to yields which is a contradiction. The proof is completed.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors were supported by the Fundamental Research Funds for the Central Universities (XDJK2012C042 and SWU113048).
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