Dynamic Transition and Pattern Formation for Chemotactic Systems

The main objective of this article is to study the dynamic transition and pattern formation for chemotactic systems modeled by the Keller-Segel equations. We study chemotactic systems with either rich or moderated stimulant supplies. For the rich stimulant chemotactic system, we show that the chemotactic system always undergoes a Type-I or Type-II dynamic transition from the homogeneous state to steady state solutions. The type of transition is dictated by the sign of a non dimensional parameter $b$. For the general Keller-Segel model where the stimulant is moderately supplied, the system can undergo a dynamic transition to either steady state patterns or spatiotemporal oscillations. From the pattern formation point of view, the formation and the mechanism of both the lamella and rectangular patterns are derived.


Introduction
Chemotaxis is a remarkable phenomenon occurring in many biological individuals, which involves mobility and aggregation of the species in two aspects: one is random walking, and the other is the chemically directed movement. For example, in the slime mould Dictyostelium discoideum, the single-cell amoebae move towards regions of relatively high concentration of a chemical called cylic-AAMP which is secreted by the amoebae themselves. Many experiments demonstrate that under some properly conditions a bacterial coloy can form a rather regular pattern, which is relatively stable in certain time scale. A series of experimental results on the patterns formed by the bacteria Escherichia coli (E. coli) and Salmonella typhimurium (S. Typhimurium) were derived in [2,3], where two types of experiments were conducted: one is in semi-solid medium, and the other is in liquid medium. They showed that when the bacteria are exposed to intermediates of TCA cycle, they can form various regular patterns, typically as ringlike and sunflowerlike formations. In all these experiments, the bacteria are known to secrete aspartate, a potent chemoattractant; also see [10,1].
The most interesting work done by Budrene and Berg are the semi-solid experiments with E. coli and S. typhimurium. A high density bacteria were inoculated in a petri dish containing a uniform distribution of stimulant in the semi-solid medium, i.e. 0.24% soft agar in succinate. The stimulant provides main food source for the bacteria. In a few days, the bacteria spread out from the inoculum, eventually covering the entire surface of the dish with a stationary pattern where the higher density population is separated by regions of near zero cell density. The S. typhimurium patterns are concentric rings and are either continuous or spotted; see Figure 1.1. The E. coli patterns are more complex with symmetry between individual aggregates. A large number of patterns has been observed. The most typical forms are concentric rings, sunflower type spirals, radial stripes, radial spots and chevrons. In the process of pattern formation, the population of bacteria has gone through many generations. The liquid experiments with E. coli and S. typhimurium exhibit relatively simple patterns which appear quickly in a few minutes, and last about half an hour before disappearing. Two types of patterns are observed, and they rely on the initial conditions. The simplest patterns are produced when the liquid medium contains a uniform distribution of bacteria and a small amount of the TCA cycle intermediate. The bacteria collect in aggregates of about the same size over the entire surface of the liquid. The second type of patterns appears when a small amount of TCA is added locally to a special spot in a uniform distribution of bacteria. In this case, the bacteria begin to form aggregates which occur on a ring centered about the special spot, and in a random arrangement inside the ring. In particular, in these liquid experiments, the timescale to form patterns is less than the time required for bacterial birth and death. Therefore, the growth of bacteria does not contribute to the pattern formation process.
Here we have to address that in these experiments, none of the chemicals placed in the petri dish is a chemo-attractant. Hence, the chemoattractants, which play a crucial role in bacterial chemotaxis, are produced and secreted by the bacteria themselves.
In their pioneering work [5], E. F. Keller and L. A. Segel proposed a model in 1970, called the Keller-Segel equations, to describe the chemotactic behaviors of the slime mould amoebae. In their equations, the growth rate of amoeba cells was ignored, i.e., the model can only depict the chemotaxis process in a small timescale, as exhibited in the liquid medium experiments with E. Coli and S. Typhimurium by [2,3]. However, in the semi-solid medium experiments, the timescale of pattern formation process is long enough to accommodate many generations of bacteria. Therefore, various revised models were presented by many authors, taking into consideration the effects of the stimulant (i.e. food source) and the growth rate of population; see among others [10] and the references therein. Also, there is a vast literature on the mathematical studies for the Keller-Segel model; see among others [13,4,12,11].
The main objective of this article is to study the dynamic transition and pattern formation for chemotactic systems modeled by the Keller-Segel equations. The study is based on the dynamic transition theory developed recently by the authors.
The key philosophy for the dynamic transition theory is to search for all transition states. The stability and the basin of attraction of the transition states provide naturally the mechanism of pattern formation associated with chemotactic systems.
Another important ingredient of the dynamic transition theory is the introduction of a dynamic classification scheme of transitions, with which phase transitions are classified into three types: Type-I, Type-II and Type-III. In more mathematically intuitive terms, they are called continuous, jump and mixed transitions respectively. Basically, as the control parameter passes the critical threshold, the transition states stay in a close neighborhood of the basic state for a Type-I transition, are outside of a neighborhood of the basic state for a Type-II (jump) transition. For the Type-III transition, a neighborhood is divided into two open regions with a Type-I transition in one region, and a Type-II transition in the other region.
Two types of Keller-Segel models are addressed in this article. The first is the model for rich stimulant chemotactic systems (with rich nutrient supplies). In this case, the equations are a two-component system, describing the evolution of the population density of biological individuals and the chemoattractant concentration. In this case we show that the chemotactic system always undergoes a Type-I or Type-II dynamic transition from the homogeneous state to steady state solutions. The type of transition is dictated by the sign of a nondimensional parameter b. For example, in a a non-growth system in a narrow domain, for the spatial scale smaller than a critical number, the system undergoes a Type-I (continuous) transition, otherwise the system undergoes a Type-II (jump) transition, leading to a more complex pattern away from the basic homogeneous state.
The second is a more general Keller-Segel model where the stimulant is moderately supplied. In this case, the model is a three-component system describing the evolution of the population density of biological individuals, the chemoattractant concentration, and the stimulant concentration. In this case, the system can undergo a dynamic transition to either steady state patterns or spatiotemporal oscillation. In both cases, the transition can be either a Type-I or Type-II dictated respectively by two nondimensional parameter b 0 and b 1 .
For simplicity, we consider in this article only the case where the first eigenvalue of the linearized problem around the homogeneous pattern is simple (real or complex), and we shall explore more general case elsewhere. In the case considered, for the Type-I transition, when the linearized eigenvalue is simple, we show that both the lamella and rectangular pattern can form depending on the geometry of the spatial domain. Namely, for narrow domains, the lamella pattern forms, otherwise the rectangular pattern occurs. For Of course, for Type-II transitions, more complex patterns emerge far from the basic homogeneous state.
The paper is arranged as follows. Section 2 introduces the Keller segel model. The rich stimulant case is addressed in Section 3, and the general there-component system is studied in Section 4. Section 5 explores some biological conclusions of the main theorems.

Keller-Segel Model
The general form of the revised Keller-Segel model is given by where u 1 is the population density of biological individuals, u 2 is the chemoattractant concentration, u 3 is the stimulant concentration, q(x) is the nutrient source, and χ is a chemotactic response coefficient. Equations (2.1) are supplemented with the Neumann condition: For simplicity, we consider in this article the case where the spatial domain Ω is a two-dimensional (2D) rectangle: It is convenient to introduce the nondimensional form of the model. For this purpose, let and we define the following non dimensional parameters: Then suppressing the primes, the non-dimensional form of the Keller-Segel model is given by: The non-dimensional of Ω is written as Often times, the following form of the Keller-Segel equations is discussed in some literatures: The biological significance of (2.6) is that the diffusion and degradation of the chemoattractant secreted by the bacteria themselves are almost balanced by their production. The main advantage of (2.6) lies in its mathematical simplicity, and as we shall see from the main results of this article, the the main characteristics of the pattern formation associated with the model are retained.

Dynamic Transitions for Rich Stimulant System
3.1. The model. We know that as nutrient u 3 is richly supplied, the Keller-Segel model (2.1) is reduced to a two-component system: It is easy to see that u * = (1, λ) is a steady state of (3.1). Consider the deviation from u * : Suppressing the primes, the system (3.1) is then transformed into 3.2. Dynamic transition and pattern formation for the diffusion and degradation balanced case. We start with an important case where the diffusion and degradation of the chemoattractant secreted by the bacteria themselves are almost balanced by their production. In this case, the second equation in (3.2) is given by With the Newman boundary condition for u 2 , we have u 2 = [−△ + 1] −1 u 1 and the functional form of the resulting equations are given by where the operators L λ : H 1 → H and G : Here the two Hilbert spaces H and H 1 are defined by To study the dynamic transition of this problem, we need to consider the linearized eigenvalue problem of (3.3): Let ρ k and e k be the eigenvalues and eigenfunctions of −∆ with the Neumann boundary condition given by for any k = (k 1 , k 2 ) ∈ N 2 + . Here N + is the set of all nonnegative integers. In particular, e 0 = 1 and ρ 0 = 0.
Obviously, the functions in (3.6) are also eigenvectors of (3.5), and the corresponding eigenvalues β k are Define a critical parameter by Let S = K = (K 1 , K 2 ) ∈ N 2 + achieves the minimization in (3.8) . Then it follows from (3.7) and (3.8) that Notice that for any K = (K 1 , K 2 ) ∈ S, K = 0, and We note that for properly choosing spatial geometry, we have Conditions (3.9) and (3.10) give rise to a dynamic transition of (3.3) from (u, λ) = (0, λ c ). For simplicity, we denote The following is the main dynamic transition theorem, providing a precise criterion for the transition type and the pattern formation mechanism of the system. Theorem 3.1. Let b be the parameter defined by (3.14). Assume that the eigenvalue β k satisfying (3.9) is simple. Then, for the system (3.3) we have the following assertions: (1) The system always undergoes a dynamic transition at (u, λ) = (0, λ c ). Namely, the basic state u = 0 is asymptotically stable for λ < λ c , and is unstable for λ > λ c . (2) For the case where b < 0, this transition is continuous (Type-I). I particular, the system bifurcates from (0, λ c ) to two steady state solutions on λ > λ c , which can be expressed as For the case b > 0, this transition is jump (Type-II), and the system has two saddle-node bifurcation solutions at some λ * (0 < λ * < λ c ) such that there are two branches v λ 1 and v λ 2 of steady states bifurcated from (v * , λ * ), and there are two other branches v λ 3 and v λ 4 bifurcated from (u * , λ * ). In Two remarks are now in order.
Remark 3.1. From the pattern formation point of view, for the Type-I transition, the patterns described by the transition solutions given in (3.15) are either lamella or rectangular: In the case where b > 0, the system undergoes a more drastic change. As λ * < λ < λ c , the homogeneous state, the new patterns v λ 2 and v λ 4 are metastable. For λ > λ c , the system undergoes transitions to more complex patterns away from the basic homogeneous sate form.
3), then Theorem 3.1 still holds true except the assertion on the existence of the two saddle-node bifurcation solutions, and the parameter should be replaced by Pattern formation and dynamic transition for the general case. Consider the general case (3.1). In this case, the unknown variable becomes u = (u 1 , u 2 ), and the basic function spaces are then defined by The linearized eigenvalue problem of (3.2) is where L λ : H 1 → H is defined by (3.16). Let B λ k be the matrices given by where ρ k are the eigenvalues as in (3.6). It is easy to see that all eigenvectors ϕ k and eigenvalues β k of (3.17) can be expressed as follows where e k are as in (3.6), and β k are also the eigenvalues of B λ k . By (3.18), β k can be expressed by

DYNAMIC TRANSITION AND PATTERN FORMATION FOR CHEMOTACTIC SYSTEMS 9
Let λ c be the parameter as defined by (3.8). It follows from (3.21) and (3.8) that Then we have the following dynamic transition theorem.
Theorem 3.2. Let b be the parameter defined by (3.14). Assume that the eigenvalue β + K satisfying (3.22) is simple. Then the assertions of Theorem 3.1 hold true for (3.2), with the expression (3.15) replaced by

Proof of Main Theorems.
Proof of Theorem 3.1. Assertion (1) follows directly from the general dynamic transition theorem in Chapter 2 of [7]. To prove Assertions (2) and (3), we need to reduce (3.3) to the center manifold near λ = λ c . We note that although the underlying system is now quasilinear in this general case, the center manifold reduction holds true as well; see [6] for details. To this end, let u = xe k + Φ, where Φ(x) the center manifold function of (3.3). Since L λ : H 1 → H is symmetric, the reduced equation is given by where G : H 1 → H is defined by (3.4), and It is known that the center manifold function satisfies that Φ(x) = O(x 2 ). A direct computation shows that It is clear that We infer from (3.26) that Using the approximation formula for center manifold functions given in (A.11) in [8], Φ satisfies the equation In view of (3.6), we find (3.29) Thus, (3.28) is written as Denote by Note that Then, by (3.11) and (3.30)-(3.32) we obtain Inserting (3.31) and (3.6) into (3.27) we get Also, we note that Then, putting (3.33) into (3.34) we get where b is the parameter given by (3.14). By (3.24) and (3.35), we derive the reduced equation on the center manifold as follows: Based on the dynamic transition theory developed in Chapter 2 in [7], we obtain Assertions (2) and (3), except that two saddle-node bifurcations occur at the same point λ = λ * . To prove this conclusion, we note that if u * (x) is a steady state solution of (3.3), then is also a steady state solution of (3.3). This is because the eigenvectors (3.6) form an orthogonal base of H 1 . Hence, two saddle-node bifurcations on λ < λ c imply that they must occur at the same point λ = λ * . Thus the proof of the theorem is complete.
Proof of Theorem 3.2. Assertion (1) follows from (3.22) and (3.23). To prove Assertions (2) and (3), we need to get the reduced equation of (3.2) to the center manifold near λ = λ c . Let u = x · ϕ K + Φ, where ϕ K is the eigenvector of (3.17) corresponding to β K at λ = λ c , and Φ(x) the center manifold function of (3.2). Then the reduced equation of (3.2) read Here ϕ * K is the conjugate eigenvector of ϕ K . By (3.19) and (3.20), ϕ K is written as which yields (3.42) (ξ * 1 , ξ * 2 ) = (ρ K+1 , ρ K ). By (3.16), the nonlinear operator G is It is known that the center manifold function Then, in view of (3.38) and (3.40), by direct computation we derive that Using the approximation formula for center manifold functions given in (A.11) in [8] From (3.6) we see that Thus, (3.44) is written as It is clear that where B λ k is the matrix given by (3.18

Inserting (3.47) into (3.43), by (3.40) and (3.42) we get
By definition, we have In view of (3.47)-(3.50), the reduced equation (3.37) is given by where b is the parameter as in (3.14). Then the theorem follows readily from (3.51). The proof is complete.

Transition of Three-Component Systems
4.1. The model. Hereafter δ 0 ≥ 0 is always assumed to be a constant. Hence, (2.6) has a positive constant steady state u * given by It is easy to see that u * 3 is the unique positive real root of the cubic equation Consider the translation (4.2) (u 1 , u 2 , u 3 ) → (u * 1 + u 1 , u * 2 + u 2 , u * 1 + u 1 ).
Then equations (2.6) are equivalent to (4.3) The Taylor expansion of g at u = 0 is expressed by Define the operators L λ : H 1 → H and G λ : H 1 → H by (4.5) Then the problem (4.3) takes the following the abstract form: It is known that the inverse mapping is a bounded linear operator. Therefore we have L λ : H 1 → H is a sector operator, and We note that the transition of (4.3) from u = 0 is equivalent to that of (2.6) from u = u * . Theorems 3.1 and 3.2 show that a two-component system undergoes only a dynamic transition to steady states. As we shall see, the transition for the threecomponent system (2.5) is quite different -it can undergo both steady state and spatiotemporal transitions.

4.2.
Linearized eigenvalue of (2.6). The eigenvalue equations of (2.6) at the steady state (u * 1 , u * 2 , u * 3 ) given by (4.1) in their abstract form are given by (4.7) L λ ϕ = βϕ, where L λ : H 1 → H as defined in (4.5). The explicit form of (4.7) is given by As before, let ρ k and e k be the eigenvalue and eigenvector of −△ with Neumann boundary condition given by (3.6), and let ψ k = (ψ k 1 , ψ k 3 ) = (ξ k1 e k , ξ k3 e k ). Then, it is easy to see that ψ k is an eigenvector of (4.7) provided that (ξ k1 , ξ k3 ) ∈ R 2 is an eigenvector of the matrix A λ k : The eigenvalues β k of A λ k , which are also eigenvalues of (4.7), are expressed by To derive the PES, we introduce two parameters as follows: .

12)
Re β ± k (Λ c ) < 0, ∀k ∈ Z 2 with ρ k = ρ K (4.13) (2) As λ c < Λ c , the eigenvalue β + K * (λ) is real near λ = λ c , and all of (4.9) satisfy Proof. By (4.9) we can see that β ± k (λ) are a pair of complex eigenvalues of (4.7) near some λ = λ * , and satisfy if and only if if and only if trA λ * k < 0, det A λ * k = 0 Due to the definition of λ c and Λ c , when Λ c < λ c we have (4.16) tr A Λc K = 0, tr A Λc k < 0, ∀k ∈ Z 2 with ρ k = ρ K , det A Λc k > 0, ∀|k| ≥ 0, and when λ c < Λ c , It is known that the real parts of β ± k are negative at λ if and only if det A λ k > 0, tr A λ k < 0. Hence, Assertions (1) and (2) follow from (4.16) and (4.17) respectively. The theorem is proved. Theorem 4.2. Let Λ c and λ c be given by (4.10) and (4.11) respectively. Then the following assertions hold true for (4.3): (1) When Λ c < λ c , the system undergoes a dynamic transition to periodic solutions at (u, λ) = (0, Λ c ). In particular, if the eigenvalues β ± K satisfying (4.12) are complex simple, then there is a parameter b 0 such that the dynamic transition is continuous (Type-I) as b 0 < 0, and is jump (Type-II) as b 0 > 0 with a singularity separation of periodic solutions at some λ * < Λ c .
Remark 4.1. By applying the standard procedure used in the preceding sections, we can derive explicit formulas for the two parameters b 0 and b 1 in Theorem 4.2. However, due to their complexity, we omit the details. Instead in the following, we shall give a method to calculate b 0 , and for b 1 we refer the interested readers to the proof of Theorem 3.2.

4.4.
Computational procedure of b 0 . The procedure to compute the parameter b 0 in Assertion (1) of Theorem 4.2 is divided into a few steps as follows.
Step 1. The reduced equations of (4.6) to center manifold at λ = Λ c are expressed by where ϕ and ψ are the eigenvectors of L λ at λ = Λ c , ϕ * and ψ * the conjugate eigenvectors, and L λ , G λ : H 1 → H the operators defined by (4.5), Φ is the center manifold function.
Then, by Theorem 2.4.5 in [7], the parameter b 0 in Theorem 4.2 is obtained by where a 11 , a 24 , a 12 , a 23 can be explicitly expressed in the terms in (4.20)-(4.22).

Transition for the system (2.5).
We are now in a position to discuss the transition of (2.5). With the translation (4.2), the system (2.5) is rewritten in the following form (4.32) where g(u) is as in (4.4). Here the notation u stands for three-component unknown: Then, all eigenvalues β j k (λ) and eigenvectors ψ j k of L λ satisfy with ψ j k = (ξ j k1 e k , ξ j k2 e k , ξ j k3 e k ), and e k as in (3.6), D λ k is a 3 × 3 matrix given by

5.1.
Biological significance of transition theorems. Pattern formation is one of the characteristics for bacteria chemotaxis, and is fully characterized by the dynamic transitions. Theorems 3.1-4.3 tell us that the nondimensional parameter λ, given by plays a crucial role to determine the dynamic transition and pattern formation. Actually, the key factor in (5.1) is the product of the chemotactic coefficient χ and the production rate r 1 : χr 1 , which depends on the type of bacteria. When λ is less than some critical value λ c , the uniform distribution of biological individuals is a stable state. When λ exceeds λ c , the bacteria cells aggregate to form more complex and stable patterns. As seen in (3.11), (4.10), (4.11) and (4.35), under different biological conditions, the critical parameter λ c takes different forms and values. But, a general formula for λ c is of the following type: where ρ k are taken as the eigenvalues of −∆ with the Neumann boundary condition.
In particular, for the system with rich nutrient supplies, (5.2) becomes The eigenvalues ρ k , depending on the geometry of Ω, satisfy where L is the length scale of Ω.
It implies that when the container Ω is small, the homogenous state is state and there is no pattern formation of bacteria under any biological conditions. 5.2. Spatiotemporal oscillation. Theorems 4.2 and 4.3 show that there are two critical parameters λ c and Λ c , such that if λ c < Λ c , the patterns formed by biological organisms are steady, as exhibited by many experimental results, and if Λ c < λ c a spatial-temporal oscillatory behavior takes place. For the case with rich nutrient, u * 1 = 1, u * 3 = ∞. In this situation, λ c in (4.11) is reduced to (3.8), and obviously we have that λ c < Λ c for both (4.10) and (4.35), and the dynamic transition and pattern formation are determined by Theorems 3.1 and 3.2. Hence there is no spatiotemporal oscillations for the rich nutrient case, and the time periodic oscillation of chemotaxis occurs only for the case where the nutrient is moderately supplied.
In this case, a spatial-temporal oscillation pattern are expected for λ > Λ c . 5.3. Transition types. One of the most important aspects of the study for phase transitions is to determine the transition types for a given system. The main theorems in this article provide precise information on the transition types. In all cases, types are precisely determined by the sign of some non dimensional parameters; see b, b 0 andb 1 respectively in the main theorems. Hence a global phase diagram can be obtained easily by setting the related parameter to be zero.
Moreover, under the condition (5.7), if the scale L 1 of Ω is smaller than some critical value L c (in (5.6) L c = 2 √ 5π), i.e. L 1 < L c , the continuous transition implies that there is only one high density region of bacteria to be formed, and if L 1 > L c then the jump transition expects a large number of high density regions to appear. 5.4. Pattern formation. As mentioned before, the pattern formation behavior is dictated by the dynamic transition of the system. In this article, we studied the formation of two type patterns-the lamella and the rectangular patterns, although the approach can be generalized to study the formation of other more complex patterns. For a growth system, the critical parameter λ c takes its value at some eigenvalue ρ K of −∆ for K = (K 1 , K 2 ), as shown by (3.11) and (4.11). From the pattern formation point of view, for the Type-I transition, the patterns described by the transition solutions in thee main theorems are either lamella or rectangular: lamella pattern for K 1 K 2 = 0, rectangular pattern for K 1 K 2 = 0.
In the case where b > 0, the system undergoes a more drastic change. As λ * < λ < λ c , the homogeneous state, the new patterns v λ 2 and v λ 4 are metastable. For λ > λ c , the system undergoes transitions to more complex patterns away from the basic homogeneous state.