Evolutionarily stable movement strategies in reaction–diffusion models with edge behavior

Abstract

Many types of organisms disperse through heterogeneous environments as part of their life histories. For various models of dispersal, including reaction–advection–diffusion models in continuously varying environments, it has been shown by pairwise invasibility analysis that dispersal strategies which generate an ideal free distribution are evolutionarily steady strategies (ESS, also known as evolutionarily stable strategies) and are neighborhood invader strategies (NIS). That is, populations using such strategies can both invade and resist invasion by populations using strategies that do not produce an ideal free distribution. (The ideal free distribution arises from the assumption that organisms inhabiting heterogeneous environments should move to maximize their fitness, which allows a mathematical characterization in terms of fitness equalization.) Classical reaction diffusion models assume that landscapes vary continuously. Landscape ecologists consider landscapes as mosaics of patches where individuals can make movement decisions at sharp interfaces between patches of different quality. We use a recent formulation of reaction–diffusion systems in patchy landscapes to study dispersal strategies by using methods inspired by evolutionary game theory and adaptive dynamics. Specifically, we use a version of pairwise invasibility analysis to show that in patchy environments, the behavioral strategy for movement at boundaries between different patch types that generates an ideal free distribution is both globally evolutionarily steady (ESS) and is a global neighborhood invader strategy (NIS).

Appendix

We begin the appendix with a short remark on the relation between the infinite periodic model and the restricted two-patch model. As in Sect. 4, it is convenient to write the equation for a single population in the infinite periodic case as

$$\begin{aligned} u_t = [D(x)u]_{xx} + f(x,u). \end{aligned}$$

If we assume that the two patch types have length $2l_1$ and $2l_2,$ respectively, then the parameter functions D and f are periodic in x with period $2(l_1+l_2)=2l,$ i.e., they are invariant under the translation $x\mapsto x+2l.$ Since the functions are also piecewise constant, they are also invariant under the reflection $x\mapsto 2l_1-x.$ Combining the translation invariance with the reflection invariance, we obtain a second reflection invariance under $x\mapsto 2l_1+2l-x.$ Since the coefficient functions have these symmetry properties, every steady-state solution, $u^*,$ has the same symmetry properties. The coefficients in the corresponding eigenvalue problem

$$\begin{aligned} \lambda \phi = [D(x)\phi ]_{xx} + \partial f(x,u^*)/\partial u \; \phi \end{aligned}$$

will then also have these symmetry properties. Therefore, the eigenfunctions will have these properties as well. The fixed points of the reflection symmetry are $x=l_1$ and $x=l_1+l.$ A smooth function with such a reflection symmetry must have zero slope that these fixed points. Hence $u_x=0$ for $x=l_1$ and $x=-l_2.$ In particular, every steady-state solution of the periodic problem is also a steady-state solution of the restricted problem, and the corresponding eigenfunctions are also eigenfunctions of the restricted problem. It is obvious that every solution of the restricted problem can be continued periodically to the real line with the symmetry conditions satisfied. A similar case was discussed in more detail for a discrete-time system of equations by Musgrave and Lutscher (2015).

In the remainder of this appendix, we show the existence and uniqueness and global boundedness of solutions to our model equations on the intervals $[-l_2,0]\cup [0,l_1].$ Our proof is based on semi-group theory and closely follows the proof in Cosner (1987).

We consider the reaction–diffusion system

$$\begin{aligned} {u_{i}}_{t}&= d_i {u_{i}}_{xx} + (E_i - F_i u_i - G_i v_i)u_i = d_i {u_{i}}_{xx} +h_i(u_i,v_i), \end{aligned}$$

(61)

$$\begin{aligned} {v_{i}}_{t}&= D_i {v_{i}}_{xx} + ({{\tilde{E}}}_i - {{\tilde{F}}}_i u_i - {{\tilde{G}}}_i v_i)v_i = D_i {v_{i}}_{xx} +H_i(u_i, v_i), \end{aligned}$$

(62)

for $t\ge 0$ and

$$\begin{aligned} x \in \left\{ \begin{array}{ll} {[}0, l_1], &{}\quad i=1,\\ {[}-l_2, 0],&{}\quad i=2, \end{array}\right. \end{aligned}$$

(63)

together with boundary and interface conditions

$$\begin{aligned} {u_{1}}_{x}(l_1,t)=0={u_{2}}_{x}(-l_2,t),\quad&u_1(0,t)=k u_2(0,t),\quad d_1 u_{1x}(0,t)=d_2 u_{2x}(0,t), \end{aligned}$$

(64)

$$\begin{aligned} {v_{1}}_{x}(l_1,t)=0={v_{2}}_{x}(-l_2,t),\quad&v_1(0,t)=K v_2(0,t),\quad D_1 v_{1x}(0,t)=D_2 v_{2x}(0,t). \end{aligned}$$

(65)

All parameters are assumed positive. We begin by defining the appropriate function spaces.

1.1 Set-up of the problem

We cast the problem into the form of an abstract evolution equation

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} w + Aw = F(w), \end{aligned}$$

(66)

where $w=(u,v)^T$ and $u=(u_1, u_2)^T$ and $v=(v_1, v_2)^T.$ We define operators

$$\begin{aligned} A_u \begin{pmatrix} u_1\\ u_2 \end{pmatrix} = \begin{pmatrix} -d_1 u_{1xx}\\ -d_2 u_{2xx} \end{pmatrix} \quad \mathrm{and} \quad A_v \begin{pmatrix} v_1\\ v_2 \end{pmatrix} = \begin{pmatrix} -D_1 v_{1xx}\\ -D_2 v_{2xx} \end{pmatrix}. \end{aligned}$$

(67)

Then we can write

$$\begin{aligned} Aw = \begin{pmatrix} A_u+I &{} 0 \\ 0 &{}A_v + I \end{pmatrix} \begin{pmatrix} u\\ v \end{pmatrix} = \begin{pmatrix} -d_1 u_{1xx}+u_1\\ -d_2 u_{2xx}+u_2\\ -D_1 v_{1xx}+v_1\\ -D_2 v_{2xx}+v_2 \end{pmatrix} \end{aligned}$$

(68)

and $F(w)=(F_u, F_v)^T,$ where

$$\begin{aligned} F(w)=\begin{pmatrix} F_u(w)\\ F_v(w) \end{pmatrix} = \begin{pmatrix} h_1(u_1, v_1)+ u_1\\ h_2(u_2, v_2)+ u_2\\ H_1(u_1,v_1)+ v_1\\ H_2(u_2,v_2)+ v_2 \end{pmatrix}. \end{aligned}$$

(69)

We define the following function spaces.

$$\begin{aligned} Y_u= & {} Y_v = L^2([0, l_1])\times L^2([-l_2, 0]), \quad \mathrm{and}\quad Y=Y_u\times Y_v. \end{aligned}$$

(70)

$$\begin{aligned} W_u= & {} W_v = W^{2,2}([0, l_1])\times W^{2,2}([-l_2, 0]), \quad \mathrm{and}\quad W=W_u\times W_v. \end{aligned}$$

(71)

Since we are in one space dimension, we have $W^{2,2}\hookrightarrow {\mathcal {C}}^1.$ Hence, a function $u\in W_u$ is continuously differentiable, so that we can impose the boundary and interface conditions that we want. We therefore set $X=X_u\times X_v$ with

$$\begin{aligned} X_u = \{(u_1, u_2)\in W_u \;| \; u \quad \mathrm{satisfies}\quad (64)\} \end{aligned}$$

(72)

and accordingly for $X_v$ with (64) replaced by (65).

On $Y_u$ we define the inner product

$$\begin{aligned} \langle u, z\rangle _{Y_u} = \langle u_1, z_1\rangle _{L^2([0, l_1])} + k \langle u_2, z_2\rangle _{L^2([-l_2, 0])} \end{aligned}$$

(73)

and obtain the norm

$$\begin{aligned} \Vert u\Vert ^2_{Y_u} = \Vert u_1 \Vert ^2_{L^2([0, l_1])} + k \Vert u_2 \Vert ^2_{L^2([-l_2, 0])} \end{aligned}$$

(74)

and similarly on $Y_v$ with k replaced by K. Finally, we have

$$\begin{aligned} \Vert w \Vert ^2_Y = \Vert u\Vert ^2_{Y_u} + \Vert v\Vert ^2_{Y_v}. \end{aligned}$$

(75)

1.2 The linear problem

Proposition 7.1

The linear operator A defines an analytic semigroup on Y.

Proof

We will show that A is invertible and that the closure of the numerical range is contained in $[1,\infty ).$ Then Lemma 2 in Cosner (1987) (which is a special case of Theorem V.3.2 in Kato 1966) states that the conditions for the generation of an analytic semigroup from Part 2, Section 2 in Friedman (1969) are satisfied.

We note that since the operator A is diagonal, and since $A_u$ and $A_v$ are essentially identical, it is sufficient to show the two properties for $A_u.$

We begin by calculating the numerical range of $A_u.$

$$\begin{aligned} \langle A_u u,u\rangle _{Y_u}= & {} \int _0^{l_1} (-d_1 {u_{1}}_{xx} + u_1){{\bar{u}}}_1 \mathrm{d}x + k \int _{-l_2}^0 (-d_2 {u_{2}}_{xx} + u_2){{\bar{u}}}_2 \mathrm{d}x\\= & {} -d_1 {u_{1}}_{x}{{\bar{u}}}_1|_0^{l_1} + \int _0^{l_1} d_1 {u_{1}}_{x} {{\bar{u}}_{1}}{}_{x} \mathrm{d}x - k d_2 {u_{2}}_{x}\bar{u}_2|_{-l_2}^0\\&+ \int _{-l_2}^0 k d_2 {u_{2}}_{x}{\bar{u}_{{2}}}{}_{x} \mathrm{d}x + \langle u, u\rangle _{Y_u}. \end{aligned}$$

(Note that ${{\bar{u}}}$ denotes the complex conjugate of the function u.)

By the boundary and interface conditions (64), the first and third term cancel. The two integral terms are non-negative, and therefore, we find

$$\begin{aligned} \langle A_u u,u\rangle _{Y_u}= & {} \int _0^{l_1} d_1 {u_{1}}_{x}{\bar{u}_{1}}{}_{x} \mathrm{d}x + \int _{-l_2}^0 k d_2 {u_{2}}_{x}{\bar{u}}_{{2}}{}_{x} \mathrm{d}x + \langle u, u\rangle _{Y_u}\\\ge & {} \langle u, u\rangle _{Y_u}. \end{aligned}$$

Hence, the numerical range

$$\begin{aligned} \theta (A_u) = \{\langle A_u u,u\rangle _{Y_u}\;| \; \Vert u\Vert _{Y_u}=1\} \end{aligned}$$

(76)

is contained in $[1,\infty )$ and so is its closure. The same is true for $A_v$ and therefore also for A.

Secondly, we show that $A_u$ has a bounded inverse. Consider $(\bar{f}_1, {{\bar{f}}}_2)\in Y_u.$ There exist unique functions $\tilde{u}_{1,2}$ that satisfy

$$\begin{aligned} -d_1 {\tilde{u}}_{{1}}{}_{xx} + {{\tilde{u}}}_1&= {{\bar{f}}}_1,\qquad x\in [0, l_1],\\ -d_2 {\tilde{u}}_{{2}}{}_{xx} + {{\tilde{u}}}_2&= {{\bar{f}}}_2,\qquad x\in [-l_2, 0], \end{aligned}$$

with Neumann conditions at all boundaries, i.e., ${\tilde{u}}_{{1}}{}_{x}(l_1) = {\tilde{u}}_{{1}}{}_{x}(0)= {\tilde{u}}_{{2}}{}_{x}(0)={\tilde{u}}_{{2}}{}_{x}(-l_2)=0.$ The reason is as follows. We notice that with these boundary conditions, the two equations decouple. Then each problem is an inhomogeneous boundary value problem, a special case of a regular Sturm–Liouville problem. A unique solution exists by classical methods (e.g., an explicit calculation of the Green’s function). We need to estimate the norm. Classical results (e.g., Theorem 9.27 in Renardi and Rogers 2004) give the estimate in $W^{1,2}$

$$\begin{aligned} \Vert {{\tilde{u}}}_1\Vert _{W^{1,2}([0, l_1])}\le C \Vert {{\bar{f}}}_1\Vert _{L^{2}([0, l_1])},\qquad \Vert {{\tilde{u}}}_2\Vert _{W^{1,2}([- l_2, 0])}\le C \Vert \bar{f}_2\Vert _{L^{2}([- l_2, 0])}. \end{aligned}$$

(77)

However, we need an estimate in $W^{2,2}.$ We can write the equations as

$$\begin{aligned} {\tilde{u}}_{i}{}_{xx} = \frac{1}{d_i}({{\tilde{u}}}_i-{{\bar{f}}}_i) \end{aligned}$$

and take norms on both sides to get

$$\begin{aligned} \Vert {\tilde{u}}_{i}{}_{xx}\Vert _{L^2}\le {{\tilde{C}}} (\Vert {{\tilde{u}}}_i \Vert _{L^2}+\Vert {{\bar{f}}}_i\Vert _{L^2}). \end{aligned}$$

By the previous estimate, the right hand side can be bounded by some multiple of the $L^2$-norm of the data $f_i$ alone so that we obtain the overall estimate

$$\begin{aligned} \Vert {{\tilde{u}}}_1\Vert _{W^{2,2}([0, l_1])}\le C \Vert {{\bar{f}}}_1\Vert _{L^{2}([0, l_1])},\qquad \Vert {{\tilde{u}}}_2\Vert _{W^{2,2}([-l_2, 0])}\le C \Vert \bar{f}_2\Vert _{L^{2}([-l_2, 0])}. \end{aligned}$$

(78)

We now use the same construction of functions $y_{1,2}$ in the proof of the existence of the dominant eigenvalue to obtain functions

$$\begin{aligned} u_i = {{\tilde{u}}}_i + y_i \end{aligned}$$

(79)

that satisfy the differential equations and the same norm estimates as ${{\tilde{u}}}_i$ with potentially different constants.

The same construction works for $A_v$ and therefore we have shown that A is invertible with bounded inverse. Lemma 2 in Cosner (1987) (which is a special case of Theorem V.3.2 in Kato 1966) now states that ${\mathbb {C}}\backslash [1,\infty )$ is contained in the resolvent set of A and

$$\begin{aligned} \Vert (\lambda -A)^{-1}\Vert \le \frac{1}{\mathrm{dist}(\lambda ,\overline{\theta (A)})} \end{aligned}$$

(80)

for all $\lambda$ in the resolvent set.

Denote the distance by $d=\mathrm{dist}(\lambda ,\overline{\theta (A)}).$ We want to show that there exists a constant C such that

$$\begin{aligned} d\ge \frac{1+|\lambda |}{C} \end{aligned}$$

for $\mathfrak {R}\lambda \le 0,$ so that from (80) we get the required estimate

$$\begin{aligned} \Vert (\lambda -A)^{-1}\Vert \le \frac{C}{1+|\lambda |}. \end{aligned}$$

(81)

On the semicircle $|\lambda |$ with $\mathfrak {R}\lambda \le 0,$ the function d assumes its minimum when $\lambda$ is purely imaginary. Hence, it is enough to show the inequality on the imaginary line. Hence, we need to show the existence of a constant C such that

$$\begin{aligned} \sqrt{1+z^2}\ge \frac{1+z}{C}, \qquad z\ge 0. \end{aligned}$$

The function $z\mapsto \frac{1+z^2}{(1+z)^2}$ is positive, continuous, and bounded with $f(0)=f(\infty )=1.$ Its maximum is 1 and its minimum occurs at $x=1.$ We can take C to be the inverse of the minimum of this function.

With this, we see that A satisfies the characterization to generate an analytic semigroup according to the theory developed in Friedman (1969), Part 2, Sect. 2. The statement is also available in Theorem 36.2 in Sell and You (2002) or in the book Pazy (1983). $\square$

1.3 The nonlinear problem

We now return to the nonlinear problem (66) and prove local existence of solutions. We use the following (notation adapted) time-independent version of Lemma 3 in Cosner (1987).

Proposition 7.2

Let A be a closed linear operator on a Banach space Y such that (81) holds. Suppose that F is a function on Y such that for some $0<\beta <1$ and for any $R>0,$ there exists a constant C(R) such that

$$\begin{aligned} \Vert F(A^{-\beta } p_1) - F(A^{-\beta } p_2)\Vert _Y \le C(R)\Vert p_1 -p_2\Vert _Y \end{aligned}$$

(82)

for all $p_{1,2}\in Y$ with $\Vert p_i\Vert _Y <R.$ Then for any $p_0\in {\mathcal {D}}(A)$ and each $R > \Vert A^{-\beta } p_0\Vert _Y$ there exists a $t^*>0$ such that problem (66) has a unique solution in $[0,t^*].$

From the previous section, we know that A is a closed linear operator on Y and that the norm estimate for the resolvent holds. To find an appropriate choice of $\beta ,$ we begin with the statement of Lemma 37.8 in Sell and You (2002).

Lemma 7.3

Let A be a positive, sectorial operator on $L^q(\Omega ,{\mathbb {R}}^n)$ with domain ${\mathcal {D}}(A)\hookrightarrow W^{m,q}$ for some $m\ge 1.$ Let $0<\beta \le 1.$ Then ${\mathcal {D}}(A^\beta )\mapsto W^{k,p}$ if $p\ge q,$$k\ge 0$ and $k-n/p < m\beta -n/q.$

We apply this lemma with $n=1,$$q=p=2$ and $m=2.$ Then we get that ${\mathcal {D}}(A^\beta )\mapsto W^{1,2}$ for all $1/2<\beta \le 1.$ We now fix some $\beta \in (1/2,1).$

We pick functions $p_{1,2}\in Y$ and set $q_i = A^{-\beta }p_i.$ Since $A^{-\beta }$ maps into ${\mathcal {D}}(A^\beta )$ and since by the previous lemma and our choice of $\beta ,$ we have the embedding into $W^{1,2}$ in each component, we see that $q_i$ are continuous and there is a constant $C_1(R)$ such that $\Vert q_i\Vert _\infty \le C_1(R) \Vert q_i\Vert _{W^{1,2}}\le C_1(R) \Vert A^{-\beta }\Vert \Vert p_i\Vert _Y.$

For $\nu \in [0,1]$ we define $u(\nu )=q_2 + \nu (q_1-q_2).$ The function $\nu \mapsto F(u(\nu ))$ satisfies $F(u(1))=F(q_1)$ and $F(u(0))=F(q_2).$ We apply the fundamental theorem of calculus and the chain rule to write

$$\begin{aligned} \Vert F(A^{-\beta }p_1) - F(A^{-\beta }p_2) \Vert _Y= & {} \Vert F(q_1) - F(q_2) \Vert _Y \end{aligned}$$

(83)

$$\begin{aligned}= & {} \Vert F(u(1)) - F(u(0))\Vert _Y \end{aligned}$$

(84)

$$\begin{aligned}= & {} \Vert \int _0^1 DF(u(\nu )) \frac{d}{d\nu } u(\nu )\mathrm{d}\nu \Vert _Y. \end{aligned}$$

(85)

Clearly, the derivative of u is $\frac{d}{d\nu } u(\nu )=q_1-q_2.$ Furthermore, the nonlinearity of F consists of polynomials of degree at most 2 in each component. In particular, DF consists of at most linear combinations of the functions in $q_i.$ Since $q_i$ are bounded by the above reasoning, there is an $L^\infty$-bound $C_2=C_2(R)$ on DF for $\Vert p_i\Vert \le R.$

Hence, we get the estimate

$$\begin{aligned} \Vert F(A^{-\beta }p_1) - F(A^{-\beta }p_2) \Vert _Y\le & {} C_2(R)\Vert q_1-q_2\Vert _Y \end{aligned}$$

(86)

Therefore, the proposition applies and we obtain local existence of solutions.

Proposition 7.4

Let $p_0\in {\mathcal {D}}(A)$ and denote by w(t) the unique local solution of (66) and $w(0)=p_0.$ Then $w(t)\in {\mathcal {D}}(A)$ for all $t\in [0,t^*]$ and w as well as $\mathrm{d}w/\mathrm{d}t$ are strongly continuous in $[0,t^*].$ Furthermore, $\mathrm{d}^2 w/\mathrm{d}t^2$ exists and is strongly continuous. Finally, if the initial condition as a function of x is non-negative and appropriately bounded, then so is the solution.

Proof

The analytic semigroup generated by A maps Y into ${\mathcal {D}}(A^\beta )$ for all $\beta \ge 0$ (Theorem 37.5 in Sell and You 2002). Therefore, the solution is in ${\mathcal {D}}(A),$ see also Theorem 2, in Friedman (1965). Continuity of w with respect to time follows from Theorem 2 in Friedman (1965). Higher regularity of solutions follows from the considerations following that theorem. Specifically, if the (Fréchet) derivative of $F(A^{-\beta }p)$ exists and is Lipschitz continuous, then the solution has strong first and second derivatives and they all belong to ${\mathcal {D}}(A^\beta ).$ Since F consists of quadratic terms, the derivative consists of linear terms and is therefore Lipschitz continuous.

To show positivity of solutions with non-negative, non-zero initial data, we apply the comparison principle (Proposition 3.2). To show the upper bounds, we proceed as follows. Function $h_i$ is negative for $u_i>E_i/F_i,$ independently of $v_i.$ Hence, it is sufficient to show an upper bound for solutions of the $u_i$-equations alone.

If $k E_1/F_1\ge E_2/F_2,$ we set $u_1(x,t)=E_1/F_1$ and $u_2(x,t)=ku_1.$ If $k E_1/F_1< E_2/F_2,$ we set $u_2(x,t)=E_2/F_2$ and $u_1(x,t)=u_2/k>E_1/F_1.$ In either case, we find $h_i(u_i,0)\le 0.$ Hence, we have found an upper solution and can apply the comparison principle again. $\square$

Proposition 7.5

The local solutions obtained above are global solutions, i.e., they exist for $t\in [0,\infty ).$

Proof

We pick $T_0>0.$ As in the proof of Lemma 3 in Cosner (1987) and in the proof of Theorem 1 in Bell and Cosner (1981), we need to show that for every local solution w on $[0,T_1]$ with $T_1\le T_0,$ there exists a constant $R'$ such that $\Vert Aw\Vert _Y<R'.$ Then we can choose $R>R'$ and apply the local existence result successively on $[0,t^*],$$[t^*, 2t^*],$ and so on until $T_0.$ Since $T_0$ was arbitrary, we have global existence.

To show the existence of the constant $R',$ we note that $Aw = -w_t + F(w).$ Hence, we aim to estimate

$$\begin{aligned} \Vert -w_t + F(w)\Vert _Y. \end{aligned}$$

We set

$$\begin{aligned} E(t)= & {} \frac{1}{2}\left( \Vert w\Vert ^2_Y + \Vert w_t\Vert ^2_Y\right) \nonumber \\= & {} \frac{1}{2}\left( \Vert u\Vert ^2_{Y_u}+\Vert v\Vert ^2_{Y_v} + \Vert u_t\Vert ^2_{Y_u}+ \Vert v_t\Vert ^2_{Y_v}\right) \end{aligned}$$

(87)

and calculate $E'(t).$

The first term in (87) gives

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} \frac{1}{2}\Vert u\Vert ^2_{Y_u} = -\langle u,A_u u\rangle _{Y_u} + \langle u,F_u(u,v)\rangle _{Y_u}. \end{aligned}$$

(88)

We estimate the first of these terms as we did in the calculation of the numerical range of the operator A,  see Proposition 7.1. We obtain

$$\begin{aligned} -\langle u,A_u u\rangle _{Y_u}\le -\langle u, u\rangle _{Y_u}\le 0. \end{aligned}$$

To estimate the second of these terms, we note that by the maximum principle, non-negative solutions (u, v) are $L^\infty$ bounded independent of time (see previous proposition), so that the terms $E_i-F_i u_i - G_i v_i$ are also $L^\infty$ bounded independent of time. Then we can estimate

$$\begin{aligned} \langle u,F_u(u,v)\rangle _{Y_u}= & {} \int _0^{l_1} u_1(E_1-F_1 u_1 - G_1 v_1)u_1 \mathrm{d}x \\&+ k \int _{-l_2}^0 u_2(E_2-F_2 u_2 - G_2 v_2)u_2 \mathrm{d}x \le C_1 \langle u,u\rangle _{Y_u}. \end{aligned}$$

The second term in (87) is estimated in the exact same way.

The third term in (87) consists of three terms, namely

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} \frac{1}{2}\Vert u_t\Vert ^2_{Y_u} = -\langle u_t,A_u u_t\rangle _{Y_u} + \langle u_t,D_u F_u(u,v)u_t\rangle _{Y_u}+ \langle u_t,D_v F_u(u,v)v_t\rangle _{Y_u}. \end{aligned}$$

(89)

The first of these three terms satisfies the same estimate as the corresponding term above, i.e.,

$$\begin{aligned} -\langle u_t,A_u u_t\rangle _{Y_u}\le -\langle u_t, u_t\rangle _{Y_u}\le 0. \end{aligned}$$

The second term can be estimated in a similar way as the second term above since $D_u F_u$ consists of linear polynomials. Hence, we find

$$\begin{aligned} \langle u_t,D_u F_u(u,v)u_t\rangle _{Y_u}\le C_2 \langle u_t,u_t\rangle _{Y_u}. \end{aligned}$$

The third term is slightly different. It is given by

$$\begin{aligned} \langle u_t,D_v F_u(u,v)v_t\rangle _{Y_u}= & {} \int _0^{l_1} {u_{1}}_{t}(-G_1 u_1){v_{1}}_{t} \mathrm{d}x + k \int _{-l_2}^0 {u_{2}}_{t}(-G_2 u_2){v_{2}}_t \mathrm{d}x \\\le & {} C_3\left( \int _0^{l_1} {u_{1}}_t {v_{1}}_t \mathrm{d}x + k \int _{-l_2}^0 {u_{2}}_t {v_{2}}_t \mathrm{d}x\right) \\\le & {} C_3\left( \int _0^{l_1} ({u^2_{1}}_t+{v^2_{1}}_t) \mathrm{d}x + k \int _{-l_2}^0 ({u^2_{2}}_t+{v^2_{2}}_t) \mathrm{d}x\right) \\\le & {} C_4 \left( \langle u_t, u_t\rangle _{Y_u} + \langle v_t, v_t\rangle _{Y_v}\right) . \end{aligned}$$

A similar estimate holds for the v-component.

Altogether, we obtain the estimate $E'(t)\le {\widehat{C}}E(t).$ In particular, E can grow at most exponentially in time. In particular, $\Vert w\Vert$ and $\Vert w_t\Vert$ remain bounded for any finite time. The bound on F(w) is obvious by the $L^\infty$- bound of w. Hence, we have shown that a constant $R'$ exists as required. $\square$