Stability, bifurcation and spikes of stationary solutions in a chemotaxis system with singular sensitivity and logistic source

The current paper is devoted to the study of stability, bifurcation, and spikes of positive stationary solutions of the following parabolic-elliptic chemotaxis system with singular sensitivity and logistic source on a bounded interval $[0,L]$ complemented with Neumann boundary condition,

where $u(t,x)$ and $v(t,x)$ represent the cellular density and chemical concentration at time $t$ and location $x$, $\chi>0$ is the chemotaxis sensitivity coefficient, $a>0$ is the growth rate of the cell population, $b>0$ is the self-limitation rate of the cell population, $\mu>0$ represents the degradation rate of the chemical signal substance and $\nu>0$ is the production rate of the chemical signal substance by the cell population.

Chemotaxis models are used to describe the movements of cells or living organisms in response to gradients of some chemical substances. Various chemotaxis systems, also known as Keller-Segel systems, have been widely studied since the pioneering works [16, 17] by Keller and Segel at the beginning of 1970s on the mathematical modeling of the aggregation process of Dictyostelium discoideum.

Consider the following parabolic-elliptic chemotaxis system with singular sensitivity and logistic source in general dimensional setting,

where $\Omega\subset\mathbb{R}^{N}$ is a bounded smooth domain, and $\chi,\mu$ and $\nu$ are positive constants, and $a,b$ are nonnegative constants. A considerable amount of research has been carried out on the global existence and boundedness of positive classical solutions of (1.2).

For example, in the case that $a=b=0$ and $\mu=\nu=1$, Biler in [3] proved the global existence of positive solutions of (1.2) when $\chi\leq 1$ and $N=2$, or $\chi<2/N$ and $N\geq 2$. Fujie, Winkler, and Yokota in [11] proved the boundedness of globally defined positive solutions of (1.2) when $\chi<\frac{2}{N}$ and $N\geq 2$. Fujie and Senba in [9] proved the global existence and boundedness of classical positive solutions of (1.2) for the case of $N=2$ for any $\chi>0$.

In the case that $a$ and $b$ are positive constants, when $N=2,$ Fujie, Winkler, and Yokota in [10] proved that finite-time blow-up does not occur in (1.2), and moreover, if

then any globally defined positive solution of (1.2) is bounded. Furthermore, Cao et al. in [5] proved that, if

with the constant $\eta$ depending on $\Omega$, then for any nonnegative initial data $u_{0}\in C(\bar{\Omega})$, where

and $u_{0}\not\equiv 0$ satisfying that $\int_{\Omega}u_{0}^{-1}<16\eta b|\Omega|^{2}/\chi^{2}$, the globally defined positive solution of (1.2) converges to $\big{(}\frac{a}{b},\frac{\nu}{\mu}\frac{a}{b}\big{)}$ as $t\to\infty$ exponentially (see [5, Theorem 1] for details).

Let

Recently, Kurt and Shen proved in [22] that classical solutions of (1.2) with initial functions $u_{0}\in C^{+}(\bar{\Omega})$ exist globally and stay bounded as time evolves, provided that $u_{0}$ is not too small and $a$ is large relative to $\chi$ (see [22, Theorem 1.2(3)]), which is an interesting biological phenomenon. Moreover, some qualitative properties of (1.2) have been obtained in [23]. For example, assuming that $a$ is large relative to $\chi$, it is shown in [23] that any globally defined positive solution of (1.2) is bounded above and below eventually by some positive constants independent of its initial functions provided that they are not too small.

There are still many interesting dynamical issues to be studied for (1.2). For example, whether any globally defined positive solution of (1.2) is bounded without the assumption that $a$ is large relative to $\chi$; whether chemotaxis induces non-constant positive stationary solutions and if so, what about the stability of non-constant positive stationary solutions and whether non-constant stationary solutions develop spikes as $\chi\to\infty$, etc. Some of those issues are strongly related to the stability and bifurcation of stationary solutions of (1.2).

There are many works on bifurcation and spikes of stationary solutions of various chemotaxis models. For example, local and global bifurcation of constant positive stationary solutions and existence of spiky steady states for chemotaxis models with certain regular sensitivity and with or without logistic source are studied in [15, 18, 19, 20, 21, 31, 32], etc. In particular, consider the following chemotaxis system with regular sensitivity and logistic source,

The authors of [20] proved the existence of single boundary spikes to (1.6) when $\chi$ is sufficiently large via the standard Lyapunov–Schmidt reduction. Among others, the authors of [31] carried out bifurcation analysis on (1.6) and obtained the explicit formulas of bifurcation values and small amplitude nonconstant positive solutions.

There are also some studies on bifurcation and spikes of stationary solutions of chemotaxis models with singular sensitivity but without logistic source. For example, in [24] and [34], local and global bifurcation of positive constant solutions and existence of spiky steady states are studied for parabolic-parabolic chemotaxis models with singular sensitivity and without logistic source. It is shown in [24] that positive monotone steady states exist as long as $\chi$ is larger than the first bifurcation value $\chi^{*}$, and that the cell density function forms a spike. The results in [24] apply to (1.1) with $a=b=0$. The paper [6] investigated the existence and nonlinear stability of boundary spike-layer solutions of a chemotaxis-consumption type system with logarithmic singular sensitivity in the half space, where the physical zero-flux and Dirichlet boundary conditions are prescribed.

However, there is little study on bifurcations and spiky solutions of (1.1) with $a,b>0$. Bifurcation analysis of (1.1) with $a,b>0$ is of great importance in several aspects. For example, it will show what types of bifurcations may occur in (1.1) as $\chi$ changes and what types of spiky patterns may be developed in (1.1) as $\chi\to\infty$. It will discover some intrinsic similarity and/or difference between the singular and regular chemotaxis sensitivities. Note that, considering (1.2), for a positive classical solution $(u(t,x),v(t,x))$ to exist globally and stay bounded, it requires that $u(t,x)$ does not become arbitrarily large and $v(t,x)$ (equivalently, $u(t,x)$) does not become arbitrarily small on any finite time interval. Consider the following chemotaxis model with regular sensitivity,

For a positive classical solution $(u(t,x),v(t,x))$ of (1.7) to exist globally and stay bounded, it only requires that $u(t,x)$ does not become arbitrarily large on any finite time interval. It is of great interest to explore intrinsic similarity and difference between singular and regular chemotaxis sensitivity via various analysis.

In this paper, we carry out some bifurcation analysis of (1.1) and some relevant study on stationary solutions of (1.1). In particular, we study the local stability and instability of the positive constant solution $(\frac{a}{b},\frac{\nu}{\mu}\frac{a}{b})$ of (1.1); bifurcation solutions of (1.1) from the positive constant solution $(\frac{a}{b},\frac{\nu}{\mu}\frac{a}{b})$; properties of non-constant positive stationary solutions; spikes developed by non-constant positive stationary solutions. Among others, we obtain the following results:

• (i)

  There is $\chi^{*}:=\chi^{*}(a,\mu)>0$ depending only on $a$ and $\mu$ such that when $0<\chi<\chi^{*}$, the positive constant solution $(\frac{a}{b},\frac{\nu}{\mu}\frac{a}{b})$ of (1.1) is locally exponentially stable with respect to small perturbations in $C([0,L])$, and when $\chi>\chi^{*}$, $(\frac{a}{b},\frac{\nu}{\mu}\frac{a}{b})$ is unstable (see Theorem 2.1).

• (ii)

  There are $\chi_{k}^{*}:=\chi_{k}^{*}(a,\mu)\geq\chi^{*}$ ($k=1,2,\cdots$) depending only on $a$ and $\mu$ such that $\inf_{k\geq 1}\chi_{k}^{*}=\chi_{k^{*}}^{*}=\chi^{*}$ for some ${k^{*}}\geq 1$, $\chi_{k}^{*}$ is strictly decreasing in $k\leq{k^{*}}$ and strictly increasing in $k\geq{k^{*}}$, and under some generic condition, a branch of non-constant stationary solutions of (1.1), denoted by ${\tilde{\Gamma}_{k}(\chi)}$ , bifurcates from $(\frac{a}{b},\frac{\nu}{\mu}\frac{a}{b})$ when $\chi$ passes through $\chi_{k}^{*}$ and $\chi\in I_{k}$, where $0<\chi-\chi_{k}^{*}\ll 1$ or $0<\chi_{k}^{*}-\chi\ll 1$ for $\chi\in I_{k}$ (see Theorem 3.1).

• (iii)

  The local bifurcation branches ${\tilde{\Gamma}_{k}}=\{(\chi,{\tilde{\Gamma}_{k}(\chi)})\,|\,\chi\in I_{k}\}$ can be extended to global branches ${\tilde{\mathcal{C}}_{k}}$ (see Theorem 4.1 for detail).

• (iv)

  The $u$-components of non-constant positive stationary solutions of (1.1) stay bounded in $L^{2}(0,L)$ and the $v$-components of non-constant positive stationary solutions of (1.1) stay bounded in $H^{2}(0,L)$ as $\chi\to\infty$ (see Theorem 4.3).

• (v)

  The weak limit of the $u$-components of any sequence of non-constant positive stationary solutions $\{(u(\cdot;\chi_{n}),v(\cdot;\chi_{n}))\}$ of (1.1) with $\chi=\chi_{n}(\to\infty)$ is a constant lying in $[0,\frac{a}{b}]$ (see Theorem 4.2).

• (vi)

  A notion of spikes is introduced (see Definition 5.1) and it is proved that any given sequence $\{(u(x;\chi_{n}),v(x;\chi_{n}))\}$ of non-constant positive stationary solutions of (1.1) with $\chi=\chi_{n}(\to\infty)$ develops spikes as $n\to\infty$ at those $x^{*}$ satisfying that $\liminf_{n\to\infty}u(x^{*};\chi_{n})>\frac{a}{b}$ (see Theorem 5.1) (hence a sequence $\{(u(x;\chi_{n}),v(x;\chi_{n}))\}$ of non-constant positive stationary solutions of (1.1) with $\chi=\chi_{n}(\to\infty)$ can develop multiple spikes as $n\to\infty$).

• (vii)

  Several numerical simulations are carried out and it is observed numerically that the local bifurcation branch ${\tilde{\Gamma}_{k^{*}}(\chi)}$ in (ii) extends to $\chi=\infty$ and the solutions on ${\tilde{\mathcal{C}}_{k^{*}}}$ are locally stable for $\chi\gg 1$, stay bounded, and develop spikes as $\chi\to\infty$ (see Numerical Experiment 2 in section 6 for the case $a=b=\mu=\nu=L=1$, locally stable boundary spiky solutions are observed in this case, and Numerical Experiment 4 in section 6 for the case $a=b=\mu=\nu=1$, $L=6$, locally stable spiky solutions with multiple spikes are observed in this case).

The main methods/techniques used and/or developed in this paper for the study of stability, bifurcation, and spikes of positive stationary solutions of (1.1) can be described as follows. We study local stability/instability of positive stationary solutions of (1.1) by general perturbation theory for dynamical systems/differential equations (see [12]). The investigation of local bifurcation of the constant solution $(\frac{a}{b},\frac{\nu}{\mu}\frac{a}{b})$ of (1.1) is carried out via invariant manifold theory, in particular, center manifold theory, for dynamical systems/differential equations (see [12]). We study the global extension of local bifurcation branches of $(\frac{a}{b},\frac{\nu}{\mu}\frac{a}{b})$ by the global bifurcation theory developed by Shi and Wang (see [27]). We investigate the development of spikes of non-constant positive stationary solutions of (1.1) via some important properties of positive stationary solutions of (1.1) obtained in this paper, for example, the property (v) stated in the above. This is a novel approach. The above mentioned methods/techniques would be useful for the study of bifurcation and spikes of stationary solutions in other chemotaxis models.

We would like to point out that, many people study the existence of nonconstant positive stationary solutions by applying the local bifurcation theory of Crandall-Rabinowtz (see [7]), which can also be used to study (1.1). In this paper, we employed the center manifold theory as the theoretical framework. The computations associated with this approach are elementary, and by using this method, we obtain several information simultaneously, such as the bifurcation direction and the explicit dependence of the bifurcating solutions on the parameters.

We make the following remarks on the implications of the results obtained in this paper, and some related problems, which are worthy to be studied further.

• (a)

  Some essential difference is observed between the effects of the parameters on the dynamics of (1.1) and (1.6). For example, consider (1.1) and (1.6), and treat $\chi$ as a bifurcation parameter. Let $\chi_{k}^{*}$ be as in (2.6), that is,

  $$\chi_{k}^{*}=\frac{\mu L^{2}+k^{2}\pi^{2}}{k^{2}\pi^{2}L^{2}}\cdot\frac{k^{2}\pi^{2}+aL^{2}}{\mu},\quad k=1,2,\cdots,$$

  and $\bar{\chi}_{k}^{*}$ be as in (2.10), that is,

  $${\bar{\chi}_{k}^{*}}=\frac{\mu L^{2}+k^{2}\pi^{2}}{k^{2}\pi^{2}L^{2}}\cdot\frac{b\big{(}k^{2}\pi^{2}+aL^{2}\big{)}}{a\nu},\quad k=1,2,\cdots.$$

  Let $\chi^{*}$ and $\bar{\chi}^{*}$ be as in (2.7) and (2.11), respectively, that is,

  $$\chi^{*}=\inf_{k\geq 1}\chi_{k}^{*},\quad\bar{\chi}^{*}=\inf_{k\geq 1}\bar{\chi}_{k}^{*}.$$

  Then the constant solution $(\frac{a}{b},\frac{\nu}{\mu}\frac{a}{b})$ of (1.1) (resp. (1.6)) is locally stable when $0<\chi<\chi^{*}$ (resp. $0<\chi<\bar{\chi}^{*}$) and unstable when $\chi>\chi^{*}$ (resp. $\chi>\bar{\chi}^{*}$), and pitchfork bifurcation occurs in (1.1) (resp. (1.6)) when $\chi$ passes through $\chi_{k}^{*}$ (resp. $\bar{\chi}_{k}^{*}$) (see (i), (ii) described in the above and [31]). Hence the larger $\chi^{*}$ (resp. $\bar{\chi}^{*}$), the less influence of the chemotaxis on the dynamics of (1.1) (resp. (1.6)).

• (b)

  Observe that $\chi_{k}^{*}$ and $\chi^{*}$ depend only on the parameters $a$ and $\mu$ in (1.1), and $\bar{\chi}_{k}^{*}$ and $\bar{\chi}^{*}$ depend on all the parameters $a,b,\mu,\nu$ in (1.6). Hence there is some essential difference between the influence of the chemotaxis on the dynamics of (1.1) and (1.6). Mathematically, this difference is due to the following two factors: first, $b$ is cancelled in $\frac{u}{v}$ when linearizing (1.1) at the constant solution $(\frac{a}{b},\frac{\nu}{\mu}\frac{a}{b})$; second, $\nu$ can be assumed to be $1$ by changing $u$ to $\nu u$ (after such change of variable, $b$ becomes $\frac{b}{\nu}$). Note that

  $$\lim_{a\to\infty}\chi_{k}^{*}=\infty,\quad\lim_{\mu\to 0^{+}}\chi_{k}^{*}=\infty,$$

  and

  $$\lim_{a\to 0^{+}}\bar{\chi}_{k}^{*}=\infty,\quad\lim_{b\to\infty}\bar{\chi}_{k}^{*}=\infty,\quad\lim_{\nu\to 0^{+}}\bar{\chi}_{k}^{*}=\infty.$$

  Biologically, the difference between $\chi^{*}$ and $\bar{\chi}^{*}$ can be interpreted as follows. In the chemotaxis model (1.1) with singular sensitivity, the chemotaxis has less influence on the dynamics of (1.1) if $a$ is large or $\mu$ is small, which is natural since large $a$ and small $\mu$ prevent $v$ from becoming too small as time evolves. In the chemotaxis model (1.6) with regular sensitivity, the chemotaxis has less influence on the dynamics of (1.6) if $a$ is small, $b$ is large, or $\nu$ is small, which is also natural since small $a$, large $b$, and small $\nu$ prevent $v$ from becoming too large as time evolves. The above intrinsic difference between singular and regular chemotaxis sensitivities resulting from the local bifurcation analysis matches in certain sense the difference between the following sufficient conditions for the global existence of positive classical solutions of (1.2) and (1.7): a positive classical solution $(u(t,x),v(t,x))$ of (1.2) exists globally if $a$ is large relative to $\chi$ and the initial distribution $u(0,x)$ is not too small (see [22, Theorem 1.2(3)]), and a positive classical solution $(u(t,x),v(t,x))$ of (1.7) exists globally if $b$ is large relative to $\chi$ (see [29, Theorem 2.5]).

• (c)

  Besides rigorous bifurcation analysis near the constant solution $(\frac{a}{b},\frac{\nu}{\mu}\frac{a}{b})$, in this paper, we also perform some theoretical as well as numerical analysis of non-constant stationary solutions of (1.1) for $\chi\gg 1$ and obtain several interesting results (see (iii)-(vii) described in the above). For example, it is observed that, up to a subsequence, as $\chi_{n}\to\infty$, a sequence $\{(u(\cdot;\chi_{n}),v(\cdot;\chi_{n}))\}$ of nonconstant stationary solutions of (1.1) develops spikes at any $x^{*}$ satisfying $\liminf_{n\to\infty}u(x^{*};\chi_{n})>\frac{a}{b}$. It is interesting to further study spiky solutions of (1.1) with $\chi\gg 1$, for example, to derive asymptotic expansions of spiky patterns with $\chi\gg 1$, to perform some bifurcation analysis around spiky patterns, etc. We leave such questions for future investigation. It should be pointed out that spiky patterns of logistic Keller–Segel models with regular sensitivity has been studied in [18], [20] and [21].

• (d)

  Consider the following two parabolic-parabolic chemotaxis models

  $$\begin{cases}u_{t}=u_{xx}-\chi(uv_{x})_{x}+u(a-bu),&0<x<L\cr v_{t}=v_{xx}-\mu v+\nu u,&0<x<L\cr u_{x}(t,0)=u_{x}(t,L)=v_{x}(t,0)=v_{x}(t,L)=0,\end{cases}$$

  (1.8)

  $$\begin{cases}u_{t}=u_{xx}-\chi(\frac{u}{v}v_{x})_{x}+u(a-bu),&0<x<L\cr v_{t}=v_{xx}-\mu v+\nu u,&0<x<L\cr u_{x}(t,0)=u_{x}(t,L)=v_{x}(t,0)=v_{x}(t,L)=0.\end{cases}$$

  (1.9)

  We want to mention that the system (1.8) has very rich spatial-temporal dynamics as demonstrated by the numerical studies in [25]. For example, the authors in [25] numerically demonstrated that the long time dynamics of solutions of (1.8) undergo homogeneous stationary solution, stationary spatial patterns in which multiple-peak patterns develop, spatial-temporal periodic solutions and spatial-temporal irregular solutions which describes a form of spatial-temporal chaos as $\chi$ is increased steadily (see section 5 of [25] for details).

  Based on some numerical simulations we performed, complicated dynamics is not observed in (1.9) for various parameter sets. For example, following the same simulations performed in section 5 of [25] and using the same parameter set, we did some numerical simulations for the system (1.9). We did not observe such rich dynamics. Indeed, we only observed homogeneous stationary solution and different peak patterns depending on randomised initial conditions as $\chi$ is increased steadily. Spatial-temporal periodic solutions and spatial-temporal irregular solutions are not observed in (1.9). We also used the parameter sets we used in Section 6.1 and Section 6.2 to simulate the existence of nonconstant solutions for the system (1.9). For the parameter set used in section 6.1, we observed the same phenomenon as demonstrated in section 6.1. For the parameter set in section 6.2, in addition to the phenomenon we observed in section 6.2, we also see the boundary perks. Time-dependent solutions are also not observed. The above numerical simulations indicate that the dynamics of system (1.9) is simpler than that of (1.8). It is certainly important to further explore whether (1.9) exhibits complicated dynamics or not. We also leave this question for future investigation.

The rest of this paper is organized as follows. Section 2 is devoted to the study of the local stability and instability of the constant solution $(\frac{a}{b},\frac{\nu}{\mu}\frac{a}{b})$ of (1.1). In section 3, we study local bifurcation of $(\frac{a}{b},\frac{\nu}{\mu}\frac{a}{b})$ and stability of the local bifurcation solutions. The global extension of the local bifurcation is investigated in section 4. Several important properties of non-constant positive stationary solutions of (1.1) are also obtained in section 4. Section 5 is devoted to the study of spiky stationary solutions of (1.1). In the last section, we provide some numerical analysis on the bifurcation and spiky solutions of (1.1).

In this section, we analysis the local stability and instability of the positive constant solution $(\frac{a}{b},\frac{\nu}{\mu}\frac{a}{b})$ of (1.1) by general perturbation theory.

The following is the main theorem in this section.

In this section, we investigate the local bifurcation of (1.1) when $\chi$ passes through $\chi_{k}^{*}$, where $\chi_{k}^{*}$ is given in (2.6).

To state our main results on local bifurcation solutions from the constant solution $(\frac{a}{b},\frac{\nu}{\mu}\frac{a}{b})$, we first introduce some notions. For given $k_{0}\geq 1$, let

Let