The current paper is to study, in particular, numerically, the spreading and vanishing dynamiccs of the following attraction-repulsion chemotaxis system with a free boundary and logistic source,
where in (1.1) is a positive constant, , , and () are nonnegative constants, and and satisfy the following assumption,
(H0) and are bounded functions on , and
Biological backgrounds of (1.1) are discussed in the paper (). The free boundary condition in (1.1) is also derived in  based on the consideration of “population loss” at the front which assumes that the expansion of the spreading front is evolved in a way that the average population density loss near the front is kept at a certain preferred level of the species, and for each given species in a given homogeneous environment, this preferred density level is a positive constant determined by their specific social and biological needs, and the environment.
One of the first mathematical models of chemotaxis were introduced by Keller and Segel (, ) to describe the aggregation of certain type of bacteria in 1970. Since their publications, considerable progress has been made in the analysis of various particular case of chemotaxis (Keller-Segel) model on both bounded and unbounded fixed domain (see , , , , , , , , , , , , , , , , , and the references therein). Among the fundamental problems in studying chemotaxis model are the existence of nonnegative solutions which are globally defined in time or blow up at a finite time and the asymptotic behavior of time global solutions.
Du and Lin studied the population invasion represented by Fisher-KPP free boundary problem in 2010 . The breaking difference between the asymptotic behaviours of Fisher-KPP with a free boundary and on the fixed or fixed unbounded domain is the vanishing-spreading dichotomy, which is well supported by some empirical evidences, for example, the introduction of several bird species from Europe to North America in the 1900s was successful only after many initial attempts (see ,).
Compared to the studying chemotaxis model on fixed bounded or fixed unbounded domain and the asymptotic behaviour of Fisher-KPP equation with a free boundary, the central problems in studying system (1.1) are the existence of nonnegative solutions which are globally defined in time, the vanishing-spreading dichotomy, local persistence, local stability, and so on.
To state the main results of the current paper, we first recall some theoretical results proved in  which will all be validated in our numerical simulations.
with norm , and
with norm . Define
Let (H1)- (H3) be the following standing assumptions.
Hence implies (H1). In the case , we can choose , and then and . Hence (H1) becomes , (H2) becomes , and (H3) becomes . In the case , we can also choose , and then and . Hence (H1) (resp.(H2), (H3)) becomes . Biologically, (H1), (H2), and (H3) indicate that the chemo-attraction sensitivity is relatively small with respect to logistic damping.
When (H1) holds, we put
Note that if (H2) holds, then .
with open compact topology, where the closure is taken under the open compact topology.
Global existence [2, Theorem 1.2]: If (H1) holds, then for any , and any and any function on satisfying
(1.1) has a unique globally defined solution , , , with and . Moreover,
where is a big enough constant and .
Assume (H1). For any given , and any given and satisfying (1.6), by the nonnegativity of , for all . Hence exists. Put
We say vanishing occurs if and
We say spreading occurs if and for any ,
For given , consider the following linear equation,
The second result is about the spreading and vanishing dichotomy scenario in (1.1).
For given , and and satisfying (1.6), if spreading occurs, it is interesting to know whether local uniform persistence occurs in the sense that there is a positive constant independent of the initial data such that for any ,
and whether local uniform convergence occurs in the sense that exists locally uniformly. We have the following result along this direction.
(i) (Local uniform persistence) For any given , if and (H2) holds, then for any ,
where is as in (1.5).
(ii) (Local uniform convergence) Assume that (H3) holds, and that for any , there has a unique strictly positive entire solution , . Then for any given , if , there are such that to any , , for any ,
(iii) (Local uniform convergence) Assume that (H3) holds, and that and . Then for any given , if , then for any ,
where is the unique strictly positive entire solution of the ODE
Biologically, the invasion or spreading of the population is depending on the initial solution, initial habitat, the moving speed (). When the spreading happens, the local persistence and convergence can be guaranteed in Fisher-KPP equation with a free boundary, furthermore, there is a asymptotic spreading speed such that (see , ).
Compared to the vanishing-spreading dichotomy in Fisher-KPP equation with a free boundary, the chemotaxis system (1.1) do not have comparison principle which leads the following interesting open problems but has positive answers in Fisher-KPP case.
For given and , whether there is such that for , vanishing occurs, and for , spreading occurs.
For given , , and , whether there is such that for with , vanishing occurs, and for with , spreading occurs.
Whether there is a spreading speed such that as long as the spreading occurs.
Vanishing-spreading result [2, Theorem 1.2] indicate there is a separating value , which is independent of the chemotactic sensitivity coefficients , such that in the vanishing scenario the limiting moving boundary and when the initial habitat the spreading guaranteed. The dependence of the dynamics of the system on the chemotactic sensitivity coefficients is another important and interesting questions , . We also have the following question in this direction.
If the asymptotic spreading speed exist, whether the limit depends on the chemotactic sensitivity coefficient and .
The objective of the current paper is to study the numerical effect of the parameters on the vanishing and spreading dynamics in the system (1.1) which will give us directions in the theoretical studies. For simplicity, we only consider the constant logistic coefficients in the system, where . In general, it is always difficult to handle the attraction term in chemotaxis system which may lead to convection dominant in the system , , , . However, in the system (1.1), we have extra numerical challenges in efficiently and accurately handling the moving boundaries . These two challenges require us to construct a new numerical algorithm in the numerical study.
Thanks to the maximum principle in the elliptic equation and death damping coefficient in the parabolic equation, the chemoattraction term can be controlled by the magnitude of the population density which has a global bound by the suppressing of the death rate. The front-fixing method has been successfully applied to solve one dimensional free boundary problem , , ,  which changes the moving boundary to fixed domain, and is the main concern in our numerical studies of (1.1). Combined with the front-fixing method, finite difference in parabolic and finite volume method in the elliptic equations in the system (1.1), we construct a new algorithm in the numerical study, as a by-product we also obtain the consistency, monotonicity of the moving boundary, positivity of the solution and stability results.
Our numerical experiments validate the vanishing and spreading dichotomy in the numerical scheme of system (1.1) which is similar to Fisher-KPP equation with a free boundary and give evidences to our conjectures that:
For given and , there is such that for , vanishing occurs, and for , spreading occurs. Which means in order to spread to the half space , the moving speed should be large enough and otherwise the population will be extinct.
There is a spreading speed such that as long as the spreading occurs, which is independent of chemotactic sensitivity coefficient and . Chemical and are produced by the species and the density is close to zero near the spreading front. In such case the decisive effect of the spreading speed should not depend on the chemotactic sensitivity coefficients and .
The rest of this paper is organized in the following way. In section 2, we first use Landau transformation to transfer the moving boundary to a fixed domain, then we use the finite difference, finite volume, and iteration method to approximate the continuous chemotaxis system. We also prove the monotonicity of the moving boundary, the positivity and stability of the discrete solutions. In section 3, we study the numerical spreading-vanishing dichotomy in (1.1) which validates our theoretical results (Vanishing-spreading dichotomy, local persistence and convergence). Our simulations also indicate the dependence or independence of the vanishing-spreading dichotomy on parameters and so on. In section 4, some future works are briefly discussed.
2 Numerical approximation of the free boundary problem
In this section, we study the numerical approximation of system (1.1) with constant logistic coefficients . First through the well-known Landau transformation (see ), we convert (1.1) into a fixed spatial domain problem. In such a way the length of the moving boundary is included as another variable to be solved apart from the population density. Then we solve the converted new problem on the basis of finite difference and finite volume method. There is a circulation that each time when time variable increase we solve the elliptic equations first and using the forward differential method to find the solution of the parabolic equation for the new time.
From the elliptic equations in (1.1), we know that
Then we introduce the Landau transformation,
The elliptic boundary conditions are
Equation (2.3) takes the form:
Let denote and multiply it on both sides of the above equation.
Boundary conditions and Stefan condition take the form
The initial conditions in (1.1) become:
and the initial function is changed into which maintains:
The following is the process according to the theory of the finite difference method. First we consider the time and space discretization , , which means the interval (0,1) is divided into equal cells
and the mesh points , with . For abbreviation, the approximate value of can be denoted by , the approximate values of and can be denoted by and . Besides we write for the value of . Let us consider the central approximation of the spatial derivatives,
By using the forward approximation of the time derivative, we get
We mainly focus on solving the values of , the relevant results about can be obtained in a similar way. For (2.15), , we can get equations for and there are two other equations we need for the boundary. In order to achieve a higher accuracy (see ), we use the idea of finite volume method to handle the boundary conditions.
We get the equation for the left boundary:
The equation for the right boundary is similar:
By now we have equations which is enough to form a system of linear algebraic equations for . It is tridiagonal and there exists the unique solutions . Similarly, we can acquire the system of linear algebraic equations about and its corresponding solutions .
for . Because of the initial conditions (2.14), we can assume a fictitious value at the point , and then
Because of (2.13), it can also be written as:
), we can obtain a optimal error estimates for the numerical approximation. Consider (2.10), (2.11) and (2.12), we denote that