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Mathematical modeling of forest ecosystems by a reaction–diffusion–advection system: impacts of climate change and deforestation

Abstract

We present an innovative mathematical model for studying the dynamics of forest ecosystems. Our model is determined by an age-structured reaction–diffusion–advection system in which the roles of the water resource and of the atmospheric activity are considered. The model is abstract but constructed in such a manner that it can be applied to real-world forest areas; thus it allows to establish an infinite number of scenarios for testing the robustness and resilience of forest ecosystems to anthropic actions or to climate change. We establish the well-posedness of the reaction–diffusion–advection model by using the method of characteristics and by reducing the initial system to a reaction–diffusion problem. The existence and stability of stationary homogeneous and stationary heterogeneous solutions are investigated, so as to prove that the model is able to reproduce relevant equilibrium states of the forest ecosystem. We show that the model fits with the principle of almost uniform precipitation over forested areas and of exponential decrease of precipitation over deforested areas. Furthermore, we present a selection of numerical simulations for an abstract forest ecosystem, in order to analyze the stability of the steady states, to investigate the impact of anthropic perturbations such as deforestation and to explore the effects of climate change on the dynamics of the forest ecosystem.

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Correspondence to Guillaume Cantin.

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Appendix

Appendix

In this appendix, we present the proofs of Theorems 12, 3, and we briefly show numerical results which underpin the approximation of the non-stationary advection equation (1) by the stationary advection equation (2).

Proof of Theorem 1

The proof of Theorem 1 is divided into three steps. First, we use the method of characteristics to solve the stationary equation (19), so as to prove that the operator \(\psi \) is well defined. Since \(v \in L^\infty (\varOmega )\) may be discontinuous, we use the theory of ordinary differential equations of Carathéodory. In the second step, we prove that the operator \(\psi \) is continuous, using an integral Gronwall lemma. Finally, in the third step, we prove that the operator \(\psi \) is differentiable. We remark that its derivative is uniquely determined as the solution of an ODE.

First step. Let \(\big \lbrace \xi (x_0,\,s)\big \rbrace _{0 \le s \le S(x_0)}\) denote one characteristic line of the advection field \(\varvec{a}\), starting at \(x_0 \in \varGamma \) and ending at \(\xi \big (x_0,\,S(x_0)\big ) \in \partial \varOmega {\setminus } \varGamma \). We introduce the function \({\tilde{\rho }}\) defined along the latter characteristic line by \({\tilde{\rho }}(s) = \rho \circ \xi (x_0,\,s)\), for \(0 \le s \le S(x_0)\), with the lightened notation (18). Then it is seen that \({\tilde{\rho }}\) satisfies the following ODE:

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{d {\tilde{\rho }}}{ds} + \sigma {\tilde{\rho }} = \varphi ({\tilde{\rho }}){\tilde{v}}, \quad 0 < s \le S(x_0), \\ {\tilde{\rho }}(0) = m(x_0). \end{array}\right. } \end{aligned}$$
(43)

Now we introduce the function \(\theta \) defined on \([0,\,S(x_0)]\) by \(\theta (s) = {\tilde{\rho }}(s)e^{\sigma s}\). We easily prove that \(\theta \) satisfies the following ODE:

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{d \theta }{ds} = g(s,\,\theta ), \quad 0 < s \le S(x_0), \\ \theta (0) = m(x_0), \end{array}\right. } \end{aligned}$$
(44)

where g is given by

$$\begin{aligned} g(s,\,\theta ) = \varphi \big (\theta e^{-\sigma s} \big ) e^{\sigma s} {\tilde{v}}(s). \end{aligned}$$

Since \(v \in L^\infty (\varOmega )\), the function g is continuous in its second variable \(\theta \) but discontinuous in its first variable s. For that reason, we solve Eq. (44) as an ODE of Carathéodory type (see e.g. Filippov 2013, Chapter 1). We directly verify the following properties:

  • the function \(g(s,\,\theta )\) is defined in \([0,\,S(x_0)] \times {\mathbb {R}}\), continuous in \(\theta \) for almost every s;

  • for all \(\theta \), the function \(g(\cdot ,\,\theta )\) is measurable;

  • \(\left| g(s,\,\theta )\right| \le M(s)\) with \(M(s) = \varphi _0 e^{\sigma s} \left| {\tilde{v}}(s)\right| \in L^\infty \big ([0,\,S(x_0)]\big )\);

  • \(\left| g(s,\,\theta _1) - g(s,\,\theta _2)\right| \le L(s) \left| \theta _1 - \theta _2\right| \) with \(L(s) = \varphi _0 \left| {\tilde{v}}(s)\right| \in L^\infty \big ([0,\,S(x_0)]\big )\).

By virtue of Theorems 1 and 2 in Filippov (2013), Chapter 1, Eq. (44) admits a unique local solution, defined on \([0,\,{\bar{s}}]\) with \({\bar{s}} > 0\). That solution is given by

$$\begin{aligned} \theta (s) = m(x_0) + \int _0^s g\big (\tau ,\,\theta (\tau )\big )d\tau , \quad 0 \le s \le {\bar{s}}, \end{aligned}$$

and it is absolutely continuous on each compact interval included in \([0,\,{\bar{s}}]\). Next, the non-negativity of \(\theta (s)\) follows from Proposition A.17 in Smith and Thieme (2011) (note that the continuity is not necessary in the proof of that proposition). We continue by proving that \(\theta \) is global in \([0,\,S(x_0)]\). This simply follows from the differential inequality

$$\begin{aligned} \dfrac{d\theta }{ds} \le \varphi _0 \left\| v\right\| _\infty \theta , \quad s > 0, \end{aligned}$$

which leads to

$$\begin{aligned} \theta (s) \le m(x_0)e^{\varphi _0 \left\| v\right\| _\infty S(x_0)}, \quad \forall s \in [0,\,{\bar{s}}]. \end{aligned}$$

Thus we have \({\bar{s}} = S(x_0)\).

Now we are brought back to equation (43), which is solved by setting

$$\begin{aligned} {\tilde{\rho }}(s) = \theta (s) e^{-\sigma s}, \quad s \in [0,\,S(x_0)]. \end{aligned}$$

As constructed, \({\tilde{\rho }}\) is absolutely continuous on each compact interval included in \([0,\,S(x_0)]\), and uniquely determined by the representation formula

$$\begin{aligned} {\tilde{\rho }}(s) = m(x_0)e^{-\sigma s} + \int _0^s \varphi \big ({\tilde{\rho }}(\tau )\big ){\tilde{v}}(\tau )e^{-\sigma (s-\tau )}d\tau , \quad s \in [0,\,S(x_0)], \end{aligned}$$

which is the lightened expression for Eq. (21). Since Eq. (43) has been solved along each characteristic line of the advection field \(\varvec{a}\), we finally solve Eq. (19) in \(\varOmega \) by setting

$$\begin{aligned} \rho (x) = {\tilde{\rho }}\big (\zeta _1(x),\,\zeta _2(x)\big ), \quad x \in \varOmega , \end{aligned}$$

with the notation (18), where \(\zeta _1\) and \(\zeta _2\) are defined by (17). As constructed, \(\rho \) is continuous and non-negative along the characteristic lines of the advection field \(\varvec{a}\). Furthermore, the principle of continuity of the solution of Eq. (43) with respect to a variation of the initial condition (see Theorem 6 in Filippov 2013, Chapter 1) and the continuity of the advection field \(\varvec{a}\) [see Eq. (15)] guaranty that \(\rho \) is continuous in \({\bar{\varOmega }}\), thus belongs to \(L_+^\infty (\varOmega )\). We have proved that the operator \(\psi \) is well defined.

Second step. Let \(v, h \in L_+^\infty (\varOmega )\). We set \(\rho = \psi (v)\) and \(\rho ^h = \psi (v+h)\). By virtue of the representation formula (21), we have:

$$\begin{aligned} \begin{aligned}&\left| \psi (v+h)(x) - \psi (v)(x)\right| \\&\quad =\left| \int _0^s \varphi \big ({\tilde{\rho }}^h(\tau )\big )({\tilde{v}}+{\tilde{h}})(\tau )e^{-\sigma (s-\tau )}d\tau - \int _0^s \varphi \big ({\tilde{\rho }}(\tau )\big ){\tilde{v}}(\tau )e^{-\sigma (s-\tau )}d\tau \right| \\&\quad =\left| \int _0^s \Big [\varphi \big ({\tilde{\rho }}^h(\tau )\big ) - \varphi \big ({\tilde{\rho }}(\tau )\big ) \Big ]{\tilde{v}}(\tau )e^{-\sigma (s-\tau )}d\tau + \int _0^s \varphi \big ({\tilde{\rho }}^h(\tau )\big ){\tilde{h}}(\tau )e^{-\sigma (s-\tau )}d\tau \right| \\&\quad \le \left\| v\right\| _\infty \int _0^s \left| \varphi \big ({\tilde{\rho }}^h(\tau )\big ) - \varphi \big ({\tilde{\rho }}(\tau )\big )\right| d\tau + \left\| h\right\| _\infty \varphi _0 \int _0^s e^{-\sigma (s-\tau )}d\tau \\&\quad \le \left\| v\right\| _\infty \varphi _0\int _0^s \left| {\tilde{\rho }}^h(\tau ) - {\tilde{\rho }}(\tau )\right| d\tau + \left\| h\right\| _\infty \dfrac{\varphi _0}{\sigma }, \end{aligned} \end{aligned}$$

where we have used properties of the function \(\varphi \) given in (13). Now we introduce the function p defined by

$$\begin{aligned} p(s) = \left| {\tilde{\rho }}^h(s) - {\tilde{\rho }}(s)\right| , \quad s \ge 0. \end{aligned}$$

We have proved above that

$$\begin{aligned} p(s) \le \left\| v\right\| _\infty \varphi _0\int _0^s p(\tau )d\tau + \left\| h\right\| _\infty \dfrac{\varphi _0}{\sigma }. \end{aligned}$$

By virtue on the integral Gronwall lemma, we obtain

$$\begin{aligned} p(s) \le \left\| h\right\| _\infty \dfrac{\varphi _0}{\sigma }e^{\left\| v\right\| _\infty \varphi _0 s} \le \left\| h\right\| _\infty \dfrac{\varphi _0}{\sigma }e^{\left\| v\right\| _\infty \varphi _0 {\bar{S}}}, \end{aligned}$$

where we have used the uniform bound (16). We then obtain

$$\begin{aligned} \left\| \psi (v+h)-\psi (v)\right\| _\infty \le \left\| h\right\| _\infty \times \frac{\varphi _0}{\sigma }e^{\varphi _0\left\| v\right\| _\infty {\bar{S}}}, \end{aligned}$$

which proves the continuity of the operator \(\psi \).

Third step. Finally, let us introduce the operator \(L_v\) uniquely defined for all \(v \in L_+^\infty (\varOmega )\) by the integral formula

$$\begin{aligned} L_v\,{\tilde{h}}(s) = \int _0^s \varphi \big ({\tilde{\rho }}(\tau )\big ){\tilde{h}}(\tau )e^{-\sigma (s-\tau )}d\tau + \int _0^s\varphi '\big ({\tilde{\rho }}(\tau )\big ){\tilde{v}}(\tau )L_v\,{\tilde{h}}(\tau ) e^{\sigma (s-\tau )} d\tau , \end{aligned}$$

for all \(h \in L_+^\infty (\varOmega )\), where \(\rho = \psi (v)\). As constructed, the operator \(L_v\) is obviously linear. Now, let \(v, h \in L_+^\infty (\varOmega )\). We set again \(\rho = \psi (v)\) and \(\rho ^h = \psi (v+h)\), and we introduce the function p defined by

$$\begin{aligned} p(s) = \left| {\tilde{\rho }}^h(s) - {\tilde{\rho }}(s) - L_v\,h(s)\right| , \quad s \ge 0. \end{aligned}$$

We use the continuity and differentiability of the function \(\varphi \) in order to write:

$$\begin{aligned} \begin{aligned}&\varphi ({\tilde{\rho }}^h) = \varphi ({\tilde{\rho }}) + o(1) ~\text {as}~\left\| h\right\| _\infty \rightarrow 0,\\&\varphi ({\tilde{\rho }}^h) = \varphi ({\tilde{\rho }}) + \varphi '({\tilde{\rho }}) ({\tilde{\rho }}^h - {\tilde{\rho }}) + o\big (\left\| h\right\| _\infty \big ) ~\text {as}~\left\| h\right\| _\infty \rightarrow 0,\\ \end{aligned} \end{aligned}$$

which leads to, after basic computations:

$$\begin{aligned} p(s) \le \left\| v\right\| _\infty \int _0^s p(\tau )d\tau + o\big (\left\| h\right\| _\infty \big ) \times \dfrac{\left\| v\right\| _\infty +1}{\sigma }, \quad s \ge 0. \end{aligned}$$

Using the integral Gronwall lemma again and Eq. (16) leads to

$$\begin{aligned} \left| {\tilde{\rho }}^h(s) - {\tilde{\rho }}(s) - L_v\,h(s)\right| \le o\big (\left\| h\right\| _\infty \big ) \times \dfrac{\left\| v\right\| _\infty +1}{\sigma } e^{\left\| v\right\| _\infty {\bar{S}}}. \end{aligned}$$

Furthermore, we easily observe that

$$\begin{aligned} \left| L_v\,{\tilde{h}}(s)\right| \le \left\| v\right\| _\infty \int _0^s \left| L_v\,{\tilde{h}}(\tau )\right| + \dfrac{\varphi _0 \left\| h\right\| _\infty }{\sigma }. \end{aligned}$$

Using a third time the integral Gronwall lemma leads to

$$\begin{aligned} \left| L_v\,{\tilde{h}}(s)\right| \le \dfrac{\varphi _0 \left\| h\right\| _\infty }{\sigma }e^{\left\| v\right\| _\infty {\bar{S}}}. \end{aligned}$$

In this way, we have proved that \(\psi \) is differentiable in \(L^\infty (\varOmega )\). Its derivative is given by

$$\begin{aligned} D\psi (v)\,h = L_v\,h, \end{aligned}$$

for all \(v, h \in L_+^\infty (\varOmega )\) and satisfies (24).

Proof of Theorem 2

By virtue of Theorem 4.4 in Yagi (2009), it suffices to prove an estimation of the type

$$\begin{aligned} \begin{aligned}&\left\| F(U) - F({\tilde{U}})\right\| _X \\&\quad \le k\big (\left\| U\right\| _X + \left\| {\tilde{U}}\right\| _X\big ) \Big [\left\| A^\eta (U-{\tilde{U}})\right\| _X + (\left\| U\right\| _X+\left\| {\tilde{U}}\right\| _X)\left\| U-{\tilde{U}}\right\| _X \Big ], \end{aligned} \end{aligned}$$

for each U, \({\tilde{U}}\) in \({\mathcal {D}}(A^\eta )\), where k is an increasing function defined in \({\mathbb {R}}^+\) with values in \({\mathbb {R}}^+\). Thus we consider \(U = (u,\,v,\,w)^T\), \({\tilde{U}} = ({\tilde{u}},\,{\tilde{v}},\,{\tilde{w}})^T\) in \({\mathcal {D}}(A^\eta )\). We begin by estimating the norm of the difference \(\left\| F_1(U)-F_1({\tilde{U}})\right\| _\infty \). To that aim, we write:

$$\begin{aligned} \begin{aligned} \left\| F_1(U)-F_1({\tilde{U}})\right\| _\infty&\le \beta \delta \left\| w-{\tilde{w}}\right\| _\infty \\&\quad + \left\| \gamma _0(v)u - \gamma _0({\tilde{v}}){\tilde{u}}\right\| _\infty \\&\quad + \left\| \mu \big (\psi (v)\big )u - \mu \big (\psi ({\tilde{v}})\big ){\tilde{u}}\right\| _\infty . \end{aligned} \end{aligned}$$

Next, we use the expression of \(\gamma _0(v)\) given in (9):

$$\begin{aligned} \begin{aligned} \left\| \gamma _0(v)u - \gamma _0({\tilde{v}}){\tilde{u}}\right\| _\infty \le a\left\| v^2 u - {\tilde{v}}^2 {\tilde{u}}\right\| _\infty + 2ab \left\| v u - {\tilde{v}} {\tilde{u}}\right\| _\infty + (ab^2+c) \left\| u - {\tilde{u}}\right\| _\infty . \end{aligned} \end{aligned}$$

But we have

$$\begin{aligned} \begin{aligned} \left\| v u - {\tilde{v}} {\tilde{u}}\right\| _\infty&\le \left\| v u - v{\tilde{u}}\right\| _\infty + \left\| v{\tilde{u}} - {\tilde{v}} {\tilde{u}}\right\| _\infty \\&\le \left\| v\right\| _\infty \left\| u - {\tilde{u}}\right\| _\infty + \left\| {\tilde{u}}\right\| _\infty \left\| v - {\tilde{v}}\right\| _\infty \\&\le (\left\| U\right\| _X+\left\| {\tilde{U}}\right\| _X)\left\| U-{\tilde{U}}\right\| _X. \end{aligned} \end{aligned}$$

Similarly, we have:

$$\begin{aligned} \begin{aligned} \left\| v^2 u - {\tilde{v}}^2 {\tilde{u}}\right\| _\infty&\le \left\| v\right\| ^2_\infty \left\| u - {\tilde{u}}\right\| _\infty + \left\| {\tilde{u}}\right\| _\infty (\left\| v\right\| _\infty + \left\| {\tilde{v}}\right\| _\infty )\left\| v-{\tilde{v}}\right\| _\infty \\&\le (\left\| U\right\| ^2_X+\left\| {\tilde{U}}\right\| ^2_X)\left\| U-{\tilde{U}}\right\| _X. \end{aligned} \end{aligned}$$

Afterwards, we write:

$$\begin{aligned} \begin{aligned} \left\| \mu \big (\psi (v)\big )u - \mu \big (\psi ({\tilde{v}})\big ){\tilde{u}}\right\| _\infty&\le \left\| \mu \big (\psi (v)\big )u - \mu \big (\psi (v)\big ){\tilde{u}}\right\| _\infty \\&\quad + \left\| \mu \big (\psi (v)\big ){\tilde{u}} - \mu \big (\psi ({\tilde{v}})\big ){\tilde{u}}\right\| _\infty \\&\le \mu _0 \left\| u - {\tilde{u}}\right\| _\infty + \left\| {\tilde{u}}\right\| _\infty \times \mu _0 \left\| \psi (v)-\psi ({\tilde{v}})\right\| _\infty \\&\le \mu _0 \left\| u - {\tilde{u}}\right\| _\infty + \left\| {\tilde{u}}\right\| _\infty \times \left\| v-{\tilde{v}}\right\| _\infty \frac{ \mu _0 \varphi _0}{\sigma }e^{\varphi _0\left\| v\right\| _\infty {\bar{S}}}, \end{aligned} \end{aligned}$$

where we have used estimation (22) at the last step. Combining the above inequalities leads to

$$\begin{aligned} \left\| F_1(U) - F_1({\tilde{U}})\right\| _\infty \le k_1\big (\left\| U\right\| _X + \left\| {\tilde{U}}\right\| _X\big ) (\left\| U\right\| _X+\left\| {\tilde{U}}\right\| _X)\left\| U-{\tilde{U}}\right\| _X, \end{aligned}$$

where \(k_1\) is a continuous increasing function. We estimate \(\left\| F_2(U)-F_2({\tilde{U}})\right\| _\infty \) analogously.

It remains to estimate the norm of the difference \(\left\| F_3(U)-F_3({\tilde{U}})\right\| _2\). We write:

$$\begin{aligned} \begin{aligned}&\left\| \alpha \big (\psi (v)\big )v - \alpha \big (\psi ({\tilde{v}})\big ){\tilde{v}}\right\| _2 \\&\quad \le \left\| \alpha \big (\psi (v)\big )v - \alpha \big (\psi ({\tilde{v}})\big )v\right\| _2 + \left\| \alpha \big (\psi ({\tilde{v}})\big )v - \alpha \big (\psi ({\tilde{v}})\big ){\tilde{v}}\right\| _2 \\&\quad \le \left\| v\right\| _\infty \left\| \alpha \big (\psi (v)\big ) - \alpha \big (\psi ({\tilde{v}})\big )\right\| _2 + \left\| \alpha \big (\psi ({\tilde{v}})\big )\right\| _\infty \left\| v - {\tilde{v}}\right\| _2 \\&\quad \le \alpha _0 \left\| v\right\| _\infty \left\| \psi (v) - \psi ({\tilde{v}})\right\| _2 + \alpha _0 \left\| v - {\tilde{v}}\right\| _2, \end{aligned} \end{aligned}$$

where we have used the properties of the function \(\alpha \) given in (13). Now we use the embedding \(L^\infty (\varOmega ) \subset L^2(\varOmega )\) and estimation (22), which leads to

$$\begin{aligned} \begin{aligned} \left\| \alpha \big (\psi (v)\big )v - \alpha \big (\psi ({\tilde{v}})\big ){\tilde{v}}\right\| _2&\le C \left[ \left\| v\right\| _\infty \left\| v-{\tilde{v}}\right\| _\infty \frac{\varphi _0}{\sigma }e^{\varphi _0\left\| v\right\| _\infty {\bar{S}}} + \left\| v - {\tilde{v}}\right\| _2\right] , \end{aligned} \end{aligned}$$

where C denotes a positive constant. Since \(\left\| v - {\tilde{v}}\right\| _2\le C \left\| A^\eta (U - {\tilde{U}})\right\| _X\), we obtain:

$$\begin{aligned} \begin{aligned}&\left\| \alpha \big (\psi (v)\big )v - \alpha \big (\psi ({\tilde{v}})\big ){\tilde{v}}\right\| _2\\&\quad \le k_3\big (\left\| U\right\| _X + \left\| {\tilde{U}}\right\| _X\big ) \Big [\left\| A^\eta (U-{\tilde{U}})\right\| _X + (\left\| U\right\| _X+\left\| {\tilde{U}}\right\| _X)\left\| U-{\tilde{U}}\right\| _X \Big ], \end{aligned} \end{aligned}$$

where \(k_3\) denotes a continuous increasing function. Combining the above estimates leads to the desired conclusion.

Proof of Theorem 3

If \(s(0) > 0\), the spectral condition \(\text {Spec}(-{{\overline{A}}}) \subset \{z\in {\mathbb {C}}:\;\mathfrak {R}(z)\le -\varepsilon \}\) for some \(\varepsilon > 0\) follows directly from Lemma 1 and the above properties of the function s. Next, the local stability of the stationary state \(\big ({\bar{\rho }}(x),\,0,\,0,\,0\big )\) is guaranteed by the results of Yagi (2009) (Section 6.2).

If \(s(0) < 0\), if follows again from Lemma 1 that \(\text {Spec}(-{\bar{A}})\) contains a real positive value. To prove that the stationary state \(\big ({\bar{\rho }}(x),\,0,\,0,\,0\big )\) is unstable, by virtue of Corollary 5.1.6 in Henry (2006), it suffices to check that

$$\begin{aligned} \left\| F({\bar{U}} + z) - F({\bar{U}}) - F'({\bar{U}})z\right\| _X = O\big (\left\| z\right\| _X^p\big ), \end{aligned}$$
(45)

for z in a neighborhood of 0 and \(p>1\). To this end, we easily compute

$$\begin{aligned} F({\bar{U}} + z) - F({\bar{U}}) - F'({\bar{U}})z = \begin{pmatrix} -\gamma \big (z_2, \psi (z_2)\big )z_1 + \gamma \big (0, \psi (0)\big ) z_1 \\ -h\big (\psi (z_2)\big )z_2 + h\big (\psi (0)\big )z_2 \\ \alpha \big (\psi (z_2)\big )z_2 - \alpha \big (\psi (0)\big )z_2 \end{pmatrix}, \end{aligned}$$

for all \(z = (z_1,\,z_2,\,z_3)^T \in X\). Recall that \(X = L^\infty (\varOmega ) \times L^\infty (\varOmega ) \times L^2(\varOmega )\). We have

$$\begin{aligned} \begin{aligned} \left\| \gamma \big (z_2, \psi (z_2)\big )z_1 - \gamma \big (0, \psi (0)\big ) z_1\right\| _\infty&\le \left\| \Big [a(z_2-b)^2+c-(ab^2+c) \Big ]z_1\right\| _\infty \\&\quad + \left\| \Big [\mu \big (\psi (z_2)\big ) - \mu \big (\psi (0)\big )\Big ]z_1\right\| _\infty . \end{aligned} \end{aligned}$$

First, it is easily seen that

$$\begin{aligned} \left\| \Big [a(z_2-b)^2+c-(ab^2+c) \Big ]z_1\right\| _\infty = O\big (\left\| z\right\| _X^2\big ), \end{aligned}$$

as \(\left\| z\right\| _X\) tends to 0. Furthermore, since the function \(\mu \) and the operator \(\psi \) are differentiable, we have

$$\begin{aligned} \left\| \mu \big (\psi (z_2)\big ) - \mu \big (\psi (0)\big )\right\| _\infty = O\big (\left\| z\right\| _X\big ), \end{aligned}$$

from which it follows that

$$\begin{aligned} \left\| \Big [\mu \big (\psi (z_2)\big ) - \mu \big (\psi (0)\big )\Big ]z_1\right\| _\infty = O\big (\left\| z\right\| _X^2 \big ), \end{aligned}$$

as \(\left\| z\right\| _X\) tends to 0. We obtain

$$\begin{aligned} \begin{aligned} \left\| \gamma \big (z_2, \psi (z_2)\big )z_1 - \gamma \big (0, \psi (0)\big ) z_1\right\| _\infty = O \big (\left\| z\right\| _X^2) \end{aligned} \end{aligned}$$

as \(\left\| z\right\| _X\) tends to 0. Since the functions h and \(\alpha \) are similarly differentiable, it is shown analogously that

$$\begin{aligned} \left\| -h\big (\psi (z_2)\big )z_2 + h\big (\psi (0)\big )z_2\right\| _\infty = O \big (\left\| z\right\| _X^2) \end{aligned}$$

and

$$\begin{aligned} \left\| \alpha \big (\psi (z_2)\big )z_2 - \alpha \big (\psi (0)\big )z_2\right\| _2 = O \big (\left\| z\right\| _X^2), \end{aligned}$$

which proves (45) with \(p=2\). The proof is complete.

Approximation of the non-stationary advection equation (1) by the stationary advection equation (2)

In order to support the approximation of the non-stationary advection equation (1) by the stationary advection equation (2), we compute numerically the solution of the following system with stationary advection

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{rcl} \varvec{a} \cdot \nabla \rho &{}=&{} -\sigma \rho + \varphi (\rho )v, \\ \dfrac{\partial u}{\partial t} &{}=&{} \beta \delta w - \gamma (v,\,\rho )u - f u, \\ \dfrac{\partial v}{\partial t} &{}=&{} f u - h(\rho ) v, \\ \dfrac{\partial w}{\partial t} &{}=&{} d \varDelta w - \beta w + \alpha (\rho ) v, \end{array} \end{array}\right. } \end{aligned}$$
(46)

and in parallel, we compute the solution of the following system with non-stationary advection

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{rcl} \dfrac{\partial {{\tilde{\rho }}}}{\partial t} + \varvec{a} \cdot \nabla {{\tilde{\rho }}} &{}=&{} -\sigma {{\tilde{\rho }}} + \varphi ({{\tilde{\rho }}}){{\tilde{v}}}, \\ \dfrac{\partial {{\tilde{u}}}}{\partial t} &{}=&{} \beta \delta {{\tilde{w}}} - \gamma ({{\tilde{v}}},\,{{\tilde{\rho }}})u - f {{\tilde{u}}}, \\ \dfrac{\partial {{\tilde{v}}}}{\partial t} &{}=&{} f {{\tilde{u}}} - h({{\tilde{\rho }}}) {{\tilde{v}}}, \\ \dfrac{\partial {{\tilde{w}}}}{\partial t} &{}=&{} d \varDelta {{\tilde{w}}} - \beta {{\tilde{w}}} + \alpha ({{\tilde{\rho }}}) {{\tilde{v}}}, \end{array} \end{array}\right. } \end{aligned}$$
(47)

with the same initial conditions and the same parameters. In both cases, the solutions are attracted to the stationary heterogeneous solution \(\big ({{\bar{\rho }}}(x),\,0,\,0,\,0\big )\) given in Proposition 3, as depicted in the following Figs. 12, 13, 14 and 15.

Fig. 12
figure 12

Numerical results for \(\rho (x)\) and \({\tilde{\rho }}(x)\). Left: stationary advection. Right: non-stationary advection

Fig. 13
figure 13

Numerical results for u(x) and \({\tilde{u}}(x)\). Left: stationary advection. Right: non-stationary advection

Fig. 14
figure 14

Numerical results for v(x) and \({\tilde{v}}(x)\). Left: stationary advection. Right: non-stationary advection

Fig. 15
figure 15

Numerical results for w(x) and \({\tilde{w}}(x)\). Left: stationary advection. Right: non-stationary advection

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Cantin, G., Ducrot, A. & Funatsu, B.M. Mathematical modeling of forest ecosystems by a reaction–diffusion–advection system: impacts of climate change and deforestation. J. Math. Biol. 83, 66 (2021). https://doi.org/10.1007/s00285-021-01696-x

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  • DOI: https://doi.org/10.1007/s00285-021-01696-x

Keywords

  • Forest model
  • Water resource
  • Climate change
  • Reaction–diffusion–advection
  • Stability analysis

Mathematics Subject Classification

  • 92B05
  • 35K58
  • 35K65
  • 35P15