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A nonlocal and periodic reaction-diffusion-advection model of a single phytoplankton species

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Abstract

In this article, we are concerned with a nonlocal reaction-diffusion-advection model which describes the evolution of a single phytoplankton species in a eutrophic vertical water column where the species relies solely on light for its metabolism. The new feature of our modeling equation lies in that the incident light intensity and the death rate are assumed to be time periodic with a common period. We first establish a threshold type result on the global dynamics of this model in terms of the basic reproduction number \(\mathcal {R}_0\). Then we derive various characterizations of \(\mathcal {R}_0\) with respect to the vertical turbulent diffusion rate, the sinking or buoyant rate and the water column depth, respectively, which in turn give rather precise conditions to determine whether the phytoplankton persist or become extinct. Our theoretical results not only extend the existing ones for the time-independent case, but also reveal new interesting effects of the modeling parameters and the time-periodic heterogeneous environment on persistence and extinction of the phytoplankton species, and thereby suggest important implications for phytoplankton growth control.

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Acknowledgments

We would like to thank two anonymous referees for their helpful comments and suggestions, which greatly improve the presentation of the original manuscript.

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Correspondence to Xiao-Qiang Zhao.

Additional information

R. Peng’s research was partially supported by NSF of China (11271167, 11171319), the Program for New Century Excellent Talents in University (NCET-11-0995), the Priority Academic Program Development of Jiangsu Higher Education Institutions and Natural Science Fund for Distinguished Young Scholars of Jiangsu Province (BK20130002), and X.-Q. Zhao’s research was partially supported by the NSERC of Canada.

Appendix

Appendix

1.1 Appendix A.1: A general linear periodic-parabolic problem

In this subsection, we extend some results of Peng and Zhao (2015) concerning the principal eigenvalue of a linear periodic-parabolic problem to a more general case.

Throughout the subsection, let \(\Omega \subset {\mathbb {R}}^N\,(N\ge 1)\) be a bounded domain with the smooth boundary \(\partial \Omega \), and let \(\mathcal {A}=\mathcal {A}(x,t)\) given by

$$\begin{aligned} \mathcal {A}(x,t)w=-\displaystyle \mathop {\Sigma }\limits _{i,j=1}^Na_{ij}(x,t)\partial _i\partial _jw +\displaystyle \mathop {\Sigma }\limits _{i=1}^Na_i(x,t)\partial _i w+a_0(x,t)w \end{aligned}$$

be uniformly elliptic in the usual sense for each \(t\in [0,T]\). We assume that \(a_{ij}=a_{ji},\,(1\le i,\,j\le N)\) and \(a_{ij}, a_i, a_0\in C(\overline{\Omega }\times [0,T])\) are T-periodic in t.

As in Peng and Zhao (2015), we assume that the boundary \(\partial \Omega \) consists of \(\Gamma _1\) and \(\Gamma _2\) which are two disjoint open and closed subsets of \(\partial \Omega \). Define the boundary operator

$$\begin{aligned} \mathcal {B}(x)w= \left\{ \begin{array}{l@{\quad }l} w &{}\quad \text{ on } \Gamma _1,\\ \partial _\nu w+b_0(x)w &{}\quad \text{ on } \Gamma _2, \end{array} \right. \end{aligned}$$

where \(\nu \) is the unit exterior normal to \(\partial \Omega \) and the nonnegative function \(b_0\in C^{1+\theta _0}(\overline{\Omega })\) for some \(0<\theta _0<1\). We allow either \(\Gamma _1\) or \(\Gamma _2\) to be the empty set. From now on, we assume that the T-periodic function \(m\in C(\overline{\Omega }\times [0,T])\) and \(m(x,t)>0\) for \((x,t)\in \overline{\Omega }\times [0,T]\). It is well known (see, e.g., Hess 1991) that the following periodic-parabolic eigenvalue problem

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \displaystyle \mathcal {L}w:=w_t-\mathcal {A}w=\lambda m(x,t)w\ \ \ \ &{}\quad \mathrm{in}\ \Omega \times (0,T],\\ \displaystyle \mathcal {B}w=0 \ \ \ &{}\quad \mathrm{on}\ \partial \Omega \times (0,T],\\ w(x,0)=w(x,T)\ \ \ &{}\quad \mathrm{in}\ \Omega \end{array} \right. \end{aligned}$$

has a principal eigenvalue \(\lambda =\lambda _1(\mathcal {L})\in {\mathbb {R}}\), which is unique, in the sense that only such an eigenvalue corresponds to a positive eigenfunction \(\varphi \) (unique up to multiplication). Such a function \(\varphi \) is usually called a principal eigenfunction.

Definition 5.1

A function \(\overline{w}\in C^{2,1}(\overline{\Omega }\times [0,T])\) is called a supersolution of \(\mathcal {L}\) if \(\overline{w}\) satisfies

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \displaystyle \mathcal {L}\overline{w}\ge 0\ \ \ \ &{}\quad \mathrm{in}\ \Omega \times (0,T],\\ \displaystyle \mathcal {B}\overline{w}\ge 0 \ \ \ &{}\quad \mathrm{on}\ \partial \Omega \times (0,T],\\ \overline{w}(x,0)\ge \overline{w}(x,T)\ \ \ &{}\quad \mathrm{in}\ \Omega . \end{array} \right. \end{aligned}$$
(5.1)

The function \(\overline{w}\) is called a strict supersolution if it is a supersolution but not a solution. A subsolution \(\underline{w}\) is defined by reversing the inequality signs in (5.1).

Definition 5.2

We say that \(\mathcal {L}\) admits the strong maximum principle if \(w\in C^{2,1}(\overline{\Omega }\times [0,T])\) satisfying (5.1) implies \(w>0\) in \(\Omega \times [0,T]\) unless \(w\equiv 0\).

We have the equivalent characteristics for the principal eigenvalue, the strong maximum principle and a positive strict supersolution. Such a result in the elliptic operator case is well known (see, e.g., Amann and López-Gómez 1998; Du 2006). Indeed, Anton and López-Gómez (1992) already established such kind of equivalent relationships for a class of linear cooperative periodic-parabolic systems subject to zero Dirichlet boundary conditions.

Proposition 5.1

The following statements are equivalent:

  1. (i)

    \(\mathcal {L}\) admits the strong maximum principle property.

  2. (ii)

    \(\lambda _1(\mathcal {L})>0\).

  3. (iii)

    \(\mathcal {L}\) has a strict supersolution which is positive in \(\Omega \times [0,T]\).

The proof of Proposition 5.1 is the same as that of (Peng and Zhao 2015, Proposition 2.1). As an immediate consequence of Proposition 5.1, we have the following result.

Proposition 5.2

The following statements are valid:

  1. (i)

    If there is a function \(\overline{w}\in C^{2,1}(\overline{\Omega }\times [0,T])\), which is positive in \(\Omega \times [0,T]\), such that

    $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \displaystyle \mathcal {L}\overline{w}\ge \overline{\lambda }_1m(x,t)\overline{w}\ \ \ \ &{}\quad \mathrm{in}\ \Omega \times (0,T],\\ \displaystyle \mathcal {B}\overline{w}\ge 0 \ \ \ &{}\quad \mathrm{on}\ \partial \Omega \times (0,T],\\ \overline{w}(x,0)\ge \overline{w}(x,T)\ \ \ &{}\quad \mathrm{in}\ \Omega \end{array} \right. \end{aligned}$$

    for some real number \(\overline{\lambda }_1\), then \(\lambda _1(\mathcal {L})\ge \overline{\lambda }_1\) and the equality holds only when \(\overline{w}\) is a principal eigenfunction of \(\lambda _1(\mathcal {L})\).

  2. (ii)

    If there is a function \(\underline{w}\in C^{2,1}(\overline{\Omega }\times [0,T])\), which is positive in \(\Omega \times [0,T]\), such that

    $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \displaystyle \mathcal {L}\underline{w}\le \underline{\lambda }_1m(x,t)\underline{w}\ \ \ \ &{}\quad \mathrm{in}\ \Omega \times (0,T],\\ \displaystyle \mathcal {B}\underline{w}\le 0 \ \ \ &{}\quad \mathrm{on}\ \partial \Omega \times (0,T],\\ \underline{w}(x,0)\le \underline{w}(x,T)\ \ \ &{}\quad \mathrm{in}\ \Omega \end{array} \right. \end{aligned}$$

    for some real number \(\underline{\lambda }_1\), then \(\lambda _1(\mathcal {L})\le \underline{\lambda }_1\) and the equality holds only when \(\underline{w}\) is a principal eigenfunction of \(\lambda _1(\mathcal {L})\).

Proof

We only verify the assertion (i) since (ii) can be proved similarly. Suppose that \(\lambda _1(\mathcal {L})<\overline{\lambda }_1\). As \(m>0\), then there holds

$$\begin{aligned} \mathcal {L}\overline{w}-\lambda _1(\mathcal {L})m\overline{w}\ge (\overline{\lambda }_1-\lambda _1(\mathcal {L}))m\overline{w}\ge ,\,\not \equiv 0. \end{aligned}$$

This, together with Proposition 5.1, implies that \(\lambda _1(\mathcal {L}-\lambda _1(\mathcal {L})m)>0\), where \(\lambda _1(\mathcal {L}-\lambda _1(\mathcal {L})m)\) denotes the principal eigenvalue of

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \displaystyle \mathcal {L}w-\lambda _1(\mathcal {L})m(x,t)w=\lambda w\ \ \ \ &{}\quad \mathrm{in}\ \Omega \times (0,T],\\ \displaystyle \mathcal {B}w=0 \ \ \ &{}\quad \mathrm{on}\ \partial \Omega \times (0,T],\\ w(x,0)=w(x,T)\ \ \ &{}\quad \mathrm{in}\ \Omega . \end{array} \right. \end{aligned}$$

This contradicts \(\lambda _1(\mathcal {L}-\lambda _1(\mathcal {L})m)=0\) due to the uniqueness of the principal eigenvalue. \(\square \)

We now consider the linear periodic-parabolic eigenvalue problem in one space dimension:

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \varphi _t-Da(x,t)\varphi _{xx}-\alpha h(x,t)\varphi _x\\ \quad +V(x,t)\varphi =\lambda m(x,t)\varphi , &{}\quad 0<x<L, 0<t<T,\\ \displaystyle \varphi _{x}(x,t)=0, &{}\quad x=0,L,\ 0<t<T, \\ \varphi (x,0)=\varphi (x,T), &{}\quad 0<x<L, \end{array} \right. \end{aligned}$$
(5.2)

where \(D,\,\alpha ,\,L\) are constants with \(D,\,L>0\), the functions \(a,\,h\) and V are Hölder continuous and periodic in t with the same period T. Moreover, it is assumed that \(a(x,t),\,h(x,t)>0,\quad \forall (x,t)\in [0,L]\times [0,T]\). The positive constants D and \(\alpha \) stand for the diffusion and advection (or drift) coefficients, respectively.

Let us denote by \(\lambda _1^\mathcal {N}(\alpha ,D)\) the principal eigenvalue of (5.2). In Peng and Zhao (2015), for the weight function \(m(x,t)\equiv 1\), we investigated the asymptotic behaviors of the principal eigenvalue \(\lambda =\lambda _1\) as D goes to zero or infinity and \(\alpha \) goes to infinity. By modifying the arguments in Peng and Zhao (2012, 2015) slightly, we see that the following three results hold true.

Proposition 5.3

For any given \(D,\,L>0\), there holds

$$\begin{aligned} \lim _{\alpha \rightarrow \infty }\lambda _1^\mathcal {N}(\alpha ,D) ={{\int _0^T V(L,t)dt}\over {\int _0^Tm(L,t)dt}}\ \ \text{ and }\ \ \lim _{\alpha \rightarrow -\infty }\lambda _1^\mathcal {N}(\alpha ,D) ={{\int _0^T V(0,t)dt}\over {\int _0^Tm(0,t)dt}}. \end{aligned}$$

Proposition 5.4

The following statements are valid:

  1. (i)

    For any given \(L>0\) and \(\alpha >0\), we have

    $$\begin{aligned} \lim _{D\rightarrow 0}\lambda _1^\mathcal {N}(\alpha ,D)={{\int _0^T V(L,t)dt}\over {\int _0^Tm(L,t)dt}}. \end{aligned}$$
  2. (ii)

    For any given \(L>0\) and \(\alpha <0\), we have

    $$\begin{aligned} \lim _{D\rightarrow 0}\lambda _1^\mathcal {N}(\alpha ,D)={{\int _0^T V(0,t)dt}\over {\int _0^Tm(0,t)dt}}. \end{aligned}$$
  3. (iii)

    Assume that a(xt) is a positive constant. For any given \(L>0\) and \(\alpha =0\), we have

    $$\begin{aligned} \lim _{D\rightarrow 0}\lambda _1^\mathcal {N}(0,D)=\min _{x\in [0,L]}{{\int _0^T V(x,t)dt}\over {\int _0^Tm(x,t)dt}}. \end{aligned}$$

Proposition 5.5

Assume that \(a(x,t)\equiv a(t)\). Then for any given \(L>0\) and \(\alpha \in {\mathbb {R}}\), there holds

$$\begin{aligned} \lim _{D\rightarrow \infty }\lambda _1^\mathcal {N}(\alpha ,D)={{\int _0^L\int _0^T V(x,t)dtdx}\over {\int _0^L\int _0^Tm(x,t)dtdx}}. \end{aligned}$$

1.2 Appendix A.2: Proof of Lemma 2.1

The argument is similar to (Peng and Zhao 2012, Lemma 2.1), and for sake of completeness, we include it in full detail.

It is well known that the eigenvalue problem (2.2) admits a unique principal eigenvalue \(\mu _0\) and the corresponding eigenfunction \(\phi \) (unique up to multiplication) is positive. Thus, according to from the constant-variation formula, we have from (2.2) that

$$\begin{aligned} \phi (x,t)=\Gamma (t,\tau )\phi (x,\tau )+\int _\tau ^t \Gamma (t,s) {{g(I_0(s)e^{-k_0x})}\over {\mu _0}}\phi (x,s)ds. \end{aligned}$$
(5.3)

Recall that (2.1) holds and \(\phi \) is bounded on \([0,L]\times {\mathbb {R}}\). Then, sending \(\tau \rightarrow -\infty \), it follows that

$$\begin{aligned} \phi (x,t)=\int _{-\infty }^t \Gamma (t,s){{g(I_0(s)e^{-k_0x})}\over {\mu _0}} \phi (x,s)ds, \quad \forall t\in {\mathbb {R}}. \end{aligned}$$

By virtue of the definition of L, this gives \(L\phi =\mu _0\phi \).

Since \(I_0(t)\) is not strictly positive, the operator L may not be strongly positive. In the sequel, we have to employ a perturbation technique to prove \(\mathcal {R}_0=\mu _0\). For any given \(\epsilon >0\), let us define

$$\begin{aligned} L_\epsilon (\phi )(t):=\int _0^\infty \Gamma (t,t-a)g((I_0(t-a)+\epsilon )e^{-k_0\cdot })\phi (\cdot ,t-a)da. \end{aligned}$$

and its spectral radius by \(\mathcal {R}_{\epsilon ,0}=\rho (L_\epsilon )\). In the current setting, we have \(I_0(t-a)+\epsilon >0\) and so \(g((I_0(t-a)+\epsilon )e^{-k_0\cdot })>0\) on \([0,L]\times {\mathbb {R}}\). Thus, it is easily seen that \(L_\epsilon :\ C_T\rightarrow C_T\) is continuous, compact and strongly positive. As a consequence of the upper semicontinuity of the spectrum (Kato 1976, Sect. IV.3.1) and the continuity of a finite system of eigenvalues (Kato 1976, Sect. IV.3.5), there holds

$$\begin{aligned} \mathcal {R}_{\epsilon ,0}\rightarrow \mathcal {R}_0\ \ \text{ as }\ \epsilon \rightarrow 0. \end{aligned}$$
(5.4)

We now denote \(\mu _{\epsilon ,0}\) to be the unique positive principal eigenvalue of (2.2), where \(I_0(t)\) replaced by \(I_0(t)+\epsilon \), and \(\phi _{\epsilon ,0}\in C_T\) to be a positive eigenfunction. Proceeding similarly as before, one has \(L_\epsilon \phi _{\epsilon ,0}=\mu _{\epsilon ,0}\phi _{\epsilon ,0}\). Since the operator \(L_\epsilon \) is strongly positive, the celebrated Krein-Rutman theorem (see, e.g., Hess 1991, Theorem 7.2) enables us to conclude that \(\mathcal {R}_{\epsilon ,0}=\mu _{\epsilon ,0}\). Thus, we can use the continuous dependence of the principal eigenvalue on the weight function to assert that

$$\begin{aligned} \mathcal {R}_{\epsilon ,0}=\mu _{\epsilon ,0}\rightarrow \mu _0\ \ \text{ as } \epsilon \rightarrow 0, \end{aligned}$$

from which and (5.4), we deduce \(\mathcal {R}_0=\mu _0\).

1.3 Appendix A.3: Proof of Lemma 2.2

We rewrite (2.7) as the following

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \displaystyle e^{(\alpha /D)x}\vartheta _t-D(e^{(\alpha /D)x}\vartheta _x)_x=e^{(\alpha /D)x}[g(I_0(t)e^{-k_0x})\\ \displaystyle \quad -d(x,t)]\vartheta +\lambda _0e^{(\alpha /D)x}\vartheta , &{}\quad 0<x<L, 0<t<T,\\ \displaystyle \vartheta _{x}(x,t)=0, &{}\quad x=0,L,\ 0<t<T, \\ \vartheta (x,0)=\vartheta (x,T), &{}\quad 0<x<L. \end{array} \right. \nonumber \\ \end{aligned}$$
(5.5)

As before, we know that \(\lambda _0\) is also the principal eigenvalue of the adjoint problem of (5.5), that is, there exists \(\vartheta ^*\in C_T\) with \(\vartheta ^*>0\) on \([0,L]\times {\mathbb {R}}\) such that \((\lambda _0,\vartheta ^*)\) solves

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \displaystyle -e^{(\alpha /D)x}\vartheta ^*_t-D(e^{(\alpha /D)x}\vartheta ^*_x)_x=e^{(\alpha /D)x}[g(I_0(t)e^{-k_0x})\\ \displaystyle \quad -d(x,t)]\vartheta ^*+\lambda _0e^{(\alpha /D)x}\vartheta ^*, &{}\quad 0<x<L, 0<t<T,\\ \displaystyle \vartheta ^*_{x}(x,t)=0, &{}\quad x=0,L,\ 0<t<T, \\ \vartheta ^*(x,0)=\vartheta ^*(x,T), &{}\quad 0<x<L. \end{array} \right. \nonumber \\ \end{aligned}$$
(5.6)

Multiplying the equation (2.4) by \(\vartheta ^*\) and then integrating the resulting equation over \((0,L)\times (0,T)\) by parts, we deduce

$$\begin{aligned} \Big (1-{1\over {\mathcal {R}_0}}\Big )\int _0^T\int _0^Le^{(\alpha /D)x}g(I_0(t)e^{-k_0x})\varphi \vartheta ^*dxdt +\lambda _0\int _0^T\int _0^Le^{(\alpha /D)x}\varphi \vartheta ^*dxdt=0. \end{aligned}$$

Due to our assumptions (1.3) and (1.4), we have

$$\begin{aligned} \int _0^T\int _0^Le^{(\alpha /D)x}g(I_0(t)e^{-k_0x})\varphi \vartheta ^*dxdt>0. \end{aligned}$$

Therefore, \(1-{1\over {\mathcal {R}_0}}\) and \(\lambda _0\) have the opposite signs and our result follows.

1.4 Appendix A.4: Proof of Lemma 2.3

We just need to adapt the argument of (Du and Mei 2011, Lemma 4.1), and for reader’s convenience, we provide the details here. Since \(v^1(x,t)<v^2(x, t)\) for \(x\in [0, L]\) and all small \(t\ge 0\), there holds

$$\begin{aligned} z^1(x, t)<z^2(x, t)\; \quad \text {for}\;x\in (0, L]\;\text {and}\;t\ge 0\;\text {small}. \end{aligned}$$
(5.7)

We proceed with a contradiction analysis by supposing that Lemma 2.3 is not true. Then there exists a finite maximal time, denoted by \(t^*\), such that (5.7) holds for every \(t\in [0, t^*)\). Clearly \(z^1(x, t^*)\le z^2(x, t^*)\) for all \(x\in [0, L]\). We next show that

$$\begin{aligned} z^1(x, t^*)=z^2(x, t^*)\; \quad \text {for some}\; x\in (0, L]. \end{aligned}$$
(5.8)

If this claim fails to hold, we then have \(z^1(x, t^*)< z^2(x, t^*)\) for all \(x\in (0,L]\). Let

$$\begin{aligned} w(x, t)=z^2(x, t)-z^1(x, t). \end{aligned}$$

Then \(w(x, t)\ge 0\) for all \(0\le t\le t^*\) and \(0\le x\le L\).

On the other hand, by letting

$$\begin{aligned} z(x, t)=\int _0^x e^{(\alpha /D)s}u(s, t)ds, \end{aligned}$$

where u is a solution of (1.7), we notice that \(z(0, t)=0\) and z solves

$$\begin{aligned} z_t= & {} Dz_{xx}-\alpha z_x-d(x,t)z+\int _0^x g\left( (I_0(t)+\sigma )e^{-k_0s-kz(s, t)}\right) e^{(\alpha /D)s}u(s, t)ds\nonumber \\= & {} Dz_{xx}-\alpha z_x-d(x,t)z+G(k_0x+kz(x, t))\nonumber \\&-k_0k^{-1}\int _0^x g\left( (I_0(t)+\sigma )e^{-k_0s-kz(s, t)}\right) ds, \end{aligned}$$
(5.9)

where \(G(\eta )=k^{-1}\int _0^{\eta }g((I_0(t)+\sigma )e^{-\xi })\,d\xi \).

Thus, making use of (5.9), we find that

$$\begin{aligned} w_t\ge&Dw_{xx}-\alpha w_x-dw+C(x, t)w\nonumber \\&+k_0k^{-1}\int _0^x\left[ g\left( (I_0(t)+\sigma )e^{-k_0s-kz^1(s, t)}\right) -g\left( (I_0(t)+\sigma )e^{-k_0s-kz^2(s, t)}\right) \right] ds\nonumber \\ \ge&Dw_{xx}-\alpha w_x+[C(x, t)-d(x,t)]w \ \ \;\;\text {for}\; 0\le x\le L,\;t\in (0, t^*],\\ w(0,&\,t)=0, \; w(L, t)>0\;\;\text {for}\; t\in (0, t^*],\nonumber \\ w(x,&\,0)>0\quad \text {for}\; 0<x\le L,\nonumber \end{aligned}$$
(5.10)

where

$$\begin{aligned} C(x, t)=kG'(k_0x+k\theta (x, t)),\; \theta (x, t)\in \left[ z^1(x, t), z^2(x, t)\right] . \end{aligned}$$

By the strong maximum principle and Hopf boundary lemma, as applied to (5.10), we can assert that \(w(x, t)>0\) for \(t\in (0, t^*]\) and \(x\in (0, L]\), and \(w_x(0, t^*)>0\). Further, by means of the smoothness of w(xt), we see \(w_x(x, t)>0\) for all t close to \(t^*\) and x close to 0. Thanks to the fact \(w(0, t)\equiv 0\), it is clear that \(w(x, t)>0\) for \(0<x\le \delta \), \(t^*\le t\le t^*+\delta \) for some small \(\delta >0\). As \(w(x, t^*)>0\) for \(x\in [\delta , L]\), we can find \(\delta _0\in (0, \delta )\) such that \(w(x, t)>0\) for \(x\in [\delta , L]\) and \(t\in [t^*, t^*+\delta _0]\). As a result, we have \(w(x, t)>0\) for \(x\in (0, L]\) and \(t\in (0, t^*+\delta _0]\), which contradicts the definition of \(t^*\). So (5.8) is proved.

The above analysis infers that there exists \(x_0\in (0, L]\) such that \(w(x_0, t^*)=0\). If \(x_0=L\), then \(w_t(L, t^*)\le 0\). In addition, for the boundary conditions, there holds \( Dw_{xx}(L, t^*)-\alpha w_x(L, t^*)=0 \). Thus, from (5.10) we obtain

$$\begin{aligned} 0\ge w_t(L, t^*)\ge & {} k_0k^{-1}\int _0^L\left[ g\left( (I_0(t)+\sigma )e^{-k_0s-kz^1(s, t)}\right) \right. \\&\left. -g\left( (I_0(t)+\sigma )e^{-k_0s-kz^2(s, t)}\right) \right] ds. \end{aligned}$$

Since \(z^1(x, t^*)\le z^2(x, t^*)\) in [0, L], the above inequality holds only if \(z^1(x, t^*)\equiv z^2(x, t^*)\).

In view of (5.10), w(xt) is a supersolution of the problem

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \overline{w}_t=D\overline{w}_{xx}-\alpha \overline{w}_x-d(x,t)\overline{w},&{}\quad 0<x<L,\;0<t\le t^*,\\ \overline{w}(0, t)=\overline{w}(L, t)=0, &{}\quad 0<t\le t^*,\\ \overline{w}(x, 0)=w(x, 0)>0, &{}\quad 0<x<L.\end{array}\right. \end{aligned}$$

By the strong maximum principle, \(\overline{w}(x, t)>0\) for \(x\in (0, L)\) and \(0<t\le t^*\). On the other hand, by the comparison principle, we have \(w(x, t)\ge \overline{w}(x, t)\) for \(x\in (0,L)\) and \(0<t\le t^*\). Hence \(w(x, t^*)>0\) for \(x\in (0,L)\). This contradicts our earlier conclusion that \(w(x, t^*)\equiv 0\). Thus we yield \(w(L, t^*)>0\). By the strong maximum principle as applied to (5.10), clearly \(w(x, t^*)>0\) for \(x\in (0, L]\), contradicting (5.8). The proof is now complete.

1.5 Appendix A.5: Proof of Lemma 2.4

Let u(xt) be the solution of (1.7) with the initial data \(u_0\in C([0,L],\mathbb {R}_+)\). According to the strong maximum principle for parabolic equations, it follows that \(u(x,t)\ge 0\) for all \(x\in [0,L]\) and \(t\ge 0\). Without loss of generality, we can assume that \(u_0\ge ,\not \equiv 0\) since \(u (x,t)\equiv 0\) if \(u_0\equiv 0\). We now set

$$\begin{aligned} v(x,t)=e^{(-\alpha /D)x}u(x,t). \end{aligned}$$

Thus, \(v(x,t)\ge 0\) on \([0,L]\times [0,\infty )\) and solves

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \displaystyle v_t=Dv_{xx}+\alpha v_x+v[g(I(x,t))-d(x,t)], &{}\quad 0<x<L, t>0,\\ \displaystyle v_{x}(x,t)=0, &{}\quad x=0,L,\ t>0, \\ v(x,0)=u_0(x)e^{(-\alpha /D)x}=:v_0\ge ,\not \equiv 0, &{}\quad 0<x<L, \end{array} \right. \end{aligned}$$
(5.11)

where

$$\begin{aligned} I=I(x,t)=I_0(t)exp\Big (-k_0x-k_1\int _0^x e^{(\alpha /D)s}v(s,t)ds\Big ). \end{aligned}$$

For later purpose, for any given constant \(0\le \sigma \le 1\), let us consider the following auxiliary problem:

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \displaystyle v_t=Dv_{xx}+\alpha v_x+v[g(I_\sigma (x,t))-d(x,t)], &{}0<x<L, t>0,\\ \displaystyle v_{x}(x,t)=0, &{}x=0,L,\ t>0, \\ v(x,0)=2\max _{x\in [0,L]}v_0(x)>0, &{}0<x<L, \end{array} \right. \end{aligned}$$
(5.12)

with

$$\begin{aligned} I_\sigma (x,t)=(I_0(t)+\sigma )exp\left( -k_0x-k_1\int _0^x e^{(\alpha /D)s}v(s,t)ds\right) . \end{aligned}$$

Denote by \(v_\sigma \) the unique solution of (5.12). Clearly, for \(0\le \sigma \le 1\), \(v_\sigma \) satisfies

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \displaystyle (v_\sigma )_t\le D(v_\sigma )_{xx}+\alpha (v_\sigma )_x&{}\\ \quad + \left[ g\left( {\tilde{I}}e^{-k_0x-k\int _0^x e^{(\alpha /D)s}v_\sigma (s,t)ds}\right) -{\tilde{d}}\right] v_\sigma , &{}\quad 0<x<L, t>0,\\ \displaystyle (v_\sigma )_{x}(x,t)=0, &{}\quad x=0,L,\ t>0, \\ (v_\sigma )(x,0)=2\max _{x\in [0,L]}v_0(x), &{}\quad 0<x<L. \end{array} \right. \end{aligned}$$
(5.13)

In the above, \({\tilde{I}}=\max _{t\in [0,T]}I_0(t)+1>0\), and

$$\begin{aligned} {\tilde{d}}=\min \Big \{{d_*\over 2},\,\min _{(x,t)\in [0,L]\times [0,T]}d(x,t)\Big \}>0, \end{aligned}$$

where \(d_*\) is defined as in (2.12) with \(I_0(t)\) replaced by \({\tilde{I}}\).

We now let w be the unique solution to the problem

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \displaystyle w_t=Dw_{xx}+\alpha w_x+\left[ g\left( {\tilde{I}}e^{-k_0x-k\int _0^x e^{(\alpha /D)s}w(s,t)ds}\right) -{\tilde{d}}\right] w,\ \ &{}\quad 0<x<L, t>0,\\ \displaystyle w_{x}(x,t)=0, \ \ &{}\quad x=0,L,\ t>0, \\ w(x,0)=3\max _{x\in [0,L]}v_0(x), &{}\quad 0<x<L. \end{array} \right. \nonumber \\ \end{aligned}$$
(5.14)

By means of Lemma 2.3, for any given \(0<\sigma <1\), there holds

$$\begin{aligned} \int _0^x e^{(\alpha /D)s}v(s, t)ds\le & {} \int _0^x e^{(\alpha /D)s}v_\sigma (s, t)ds\\\le & {} \int _0^xe^{(\alpha /D)s}w(s, t)ds, \quad \forall (x,t)\in [0,L]\times [0,\infty ). \end{aligned}$$

This implies that

$$\begin{aligned} \int _0^L v(x, t)dx\le e^{(2|\alpha |/D)L}\int _0^Lw(x,t)dx, \quad \forall t\ge 0. \end{aligned}$$
(5.15)

In addition, for the autonomous problem (5.14), thanks to the choice of \({\tilde{d}}\), it follows from (Du and Mei 2011, Theorem 2.2) that

$$\begin{aligned} \lim \limits _{t\rightarrow \infty }w(x,t)= w^*(x) \text{ uniformly } \text{ for } x\in [0,L], \end{aligned}$$

where \(w^*\) is the unique positive steady state of system (5.14). Hence, there is a large time \(T_0>0\) such that \(w(x,t)<2\max _{x\in [0,L]}w^*(x)\) for \(t\ge T_0\). This, combined with (5.15), gives rise to

$$\begin{aligned} \int _0^L v(x, t)dx\le 2Le^{(2|\alpha |/D)L}\max _{x\in [0,L]}w^*(x),\quad \forall t\ge T_0. \end{aligned}$$
(5.16)

In view of (Alikakos 1979, Theorem 3.1) (also see Le 1997), we can use (5.16) and the Eq. (5.11) to obtain

$$\begin{aligned} v(x, t)\le C_1, \quad \forall x\in [0,L],\ t\ge T_0, \end{aligned}$$

for some positive constant \(C_1\), which is independent of \(u_0\). It then follows that

$$\begin{aligned} u(x,t)=e^{(\alpha /D)x}v(x,t)\le e^{(|\alpha |/D)L}C_1=:C_0,\quad \forall x\in [0,L],\ t\ge T_0. \end{aligned}$$

This proves the desired ultimate boundedness of solutions.

1.6 Appendix A.6: Proof of Lemma 3.2

The assertions (i) and (ii) follow from the same analysis as in (Peng and Zhao 2012, Lemma 2.3).

We now verify (iii). Let \(\lambda _0\) be defined as before. Then, in view of (Hess 1991, Lemma 15.6), we obtain

$$\begin{aligned} \lambda _0\ge -{1\over T}\int _0^T\max _{x\in [0,L]}(g(I_0(t)e^{-k_0x})-d(x,t))dt, \end{aligned}$$

and the inequality holds if and only if \(g(I_0(t)e^{-k_0x})-d(x,t)\) nontrivially depends on the spatial variable x. This, combined with Lemma 2.2, implies the assertion (iii).

Let \((\mathcal {R}_0,\varphi )\) be given as in (2.4). To prove (iv), we first notice that

$$\begin{aligned} \varphi _t-D\varphi _{xx}-\alpha \varphi _x+\varphi \max _{x\in [0,L]}d(x,t) \ge {{g(I_0(t)e^{-k_0L})}\over {\mathcal {R}_0}}\varphi , \quad \forall (x,t)\in [0,L]\times [0,T].\nonumber \\ \end{aligned}$$
(5.17)

We then consider the following eigenvalue problem

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \displaystyle w_t-Dw_{xx}-\alpha w_x+w\max _{x\in [0,L]}d(x,t)&{}\\ \quad =\displaystyle {{g(I_0(t)e^{-k_0L})}\over {\mu }}w,\ \ &{}\quad 0<x<L, 0<t<T,\\ \displaystyle w_{x}(x,t)=0, \ \ &{}\quad x=0,L,\ 0<t<T, \\ w(x,0)=w(x,T), &{}\quad 0<x<L. \end{array} \right. \end{aligned}$$
(5.18)

Due to the uniqueness of the principal eigenvalue of (5.18), one easily finds that (5.18) is equivalent to solving the ODE problem:

$$\begin{aligned} {{dw}\over {dt}}+w\max _{x\in [0,L]}d(x,t)={{g(I_0(t)e^{-k_0L})}\over {\mu }}w,\ 0<t<T;\ \ \ w(0)=w(T).\qquad \end{aligned}$$
(5.19)

A simple calculation shows that (5.19) has a positive solution \(w=w(t)\) if and only if

$$\begin{aligned} \mu ={{\int _0^Tg(I_0(t)e^{-k_0L})dt}\over {\int _0^T\max _{x\in [0,L]}d(x,t)dt}}. \end{aligned}$$

In view of Proposition 5.2, it follows from (5.17) and (5.19) that \(\mathcal {R}_0\ge \mu \), and hence,

$$\begin{aligned} \mathcal {R}_0\ge {{\int _0^Tg(I_0(t)e^{-k_0L})dt}\over {\int _0^T\max _{x\in [0,L]}d(x,t)dt}}. \end{aligned}$$

Similarly, since \((\mathcal {R}_0,\varphi )\) satisfies

$$\begin{aligned} \varphi _t-D\varphi _{xx}-\alpha \varphi _x+\varphi \min _{x\in [0,L]}d(x,t) \le {{g(I_0(t))}\over {\mathcal {R}_0}}\varphi ,\quad \forall (x,t)\in [0,L]\times [0,T], \end{aligned}$$

we deduce

$$\begin{aligned} \mathcal {R}_0\le {{\int _0^Tg(I_0(t))dt}\over {\int _0^T\min _{x\in [0,L]}d(x,t)dt}}. \end{aligned}$$

This yields (iv).

It remains to prove (v). Since \(\varphi >0\) on \([0,L]\times [0,T]\), we divide the equation (2.4) by \(\varphi \) and integrate the resulting equation over \((0,L)\times (0,T)\) to derive

$$\begin{aligned}&-D\int _0^T\int _0^L{{e^{(\alpha /D)x}\varphi _x^2}\over {\varphi ^2}}dxdt+ \int _0^T\int _0^Le^{(\alpha /D)x}d(x,t)dxdt\\&\quad ={1\over {\mathcal {R}_0}}\int _0^T\int _0^Le^{(\alpha /D)x}g(I_0(t)e^{-k_0x})dxdt, \end{aligned}$$

which therefore implies

$$\begin{aligned} \mathcal {R}_0\ge {{\int _0^T\int _0^Le^{(\alpha /D)x}g(I_0(t)e^{-k_0x})dxdt}\over {\int _0^T\int _0^L e^{(\alpha /D)x}d(x,t)dxdt}}. \end{aligned}$$

Clearly, the above equality holds if and only if

$$\begin{aligned} \int _0^T\int _0^L{{e^{(\alpha /D)x}\varphi _x^2}\over {\varphi ^2}}dxdt=0, \end{aligned}$$

that is, \(\varphi _x\equiv 0\) and equivalently, \(\varphi (x,t)\equiv \varphi (t)\). Hence, (2.3) becomes equivalent to

$$\begin{aligned} {{d\varphi }\over {dt}}+d(x,t)\varphi ={{g(I_0(t)e^{-k_0x})}\over {\mathcal {R}_0}}\varphi ,\ \ 0<t<T;\ \ \varphi (0)=\varphi (T). \end{aligned}$$
(5.20)

Then, it is easy to see that

$$\begin{aligned} \mathcal {R}_0={{\int _0^Tg(I_0(t)e^{-k_0x})dt}\over {\int _0^Td(x,t)dt}},\ \quad \forall x\in [0,L]. \end{aligned}$$

and \(d(x,t)-{{g(I_0(t)e^{-k_0x})}\over {\mathcal {R}_0}}\) depends only on the variable t. This equivalently means that \({{\int _0^Tg(I_0(t)e^{-k_0x})dt}\over {\int _0^Td(x,t)dt}}\) must be a constant and

$$\begin{aligned} {{g(I_0(t)e^{-k_0x})}\over {\int _0^Tg(I_0(t)e^{-k_0x})dt}}-{{d(x,t)}\over {\int _0^Td(x,t)dt}} \end{aligned}$$

depends only on t. The proof of (v) is thus complete.

1.7 Appendix A.7: Proof of Lemma 3.6

Let \((\mathcal {R}_0,\varphi )\) be given as in (2.4). For any given \(\epsilon >0\), due to the monotonicity of g, we can restrict \(0<L<\epsilon \) to be small enough so that

$$\begin{aligned} g(I_0(t))\ge g(I_0(t)e^{-k_0x})\ge g(I_0(t)e^{-\epsilon k_0}) \quad \text{ on } [0,L]\times [0,T], \end{aligned}$$

and

$$\begin{aligned} d(0,t)+\epsilon \ge d(x,t)\ge d(0,t)-\epsilon \quad \text{ on } [0,L]\times [0,T]. \end{aligned}$$

Hence, \((\mathcal {R}_0,\varphi )\) satisfies

$$\begin{aligned} \varphi _t-D\varphi _{xx}-\alpha \varphi _x+(d(0,t)+\epsilon )\varphi \ge {{g(I_0(t)e^{-k_0\epsilon })}\over {\mathcal {R}_0}}\varphi ,\quad \forall (x,t)\in [0,L]\times [0,T], \end{aligned}$$

and

$$\begin{aligned} \varphi _t-D\varphi _{xx}-\alpha \varphi _x+(d(0,t)-\epsilon )\varphi \le {{g(I_0(t))}\over {\mathcal {R}_0}}\varphi ,\quad \forall (x,t)\in [0,L]\times [0,T]. \end{aligned}$$

Thus, the similar analysis to that of Proposition 3.2(iv) gives rise to

$$\begin{aligned} {{\int _0^Tg(I_0(t)e^{-k_0\epsilon })dt}\over {\int _0^T(d(0,t)+\epsilon )dt}} \le \mathcal {R}_0\le {{\int _0^Tg(I_0(t))dt}\over {\int _0^T(d(0,t)-\epsilon )dt}}. \end{aligned}$$

The desired result is obtained by sending \(\epsilon \rightarrow 0\) in the above inequalities.

1.8 Appendix A.8: Proof of Lemma 3.7

By Lemma 3.1, \(\lim \limits _{L\rightarrow \infty }\mathcal {R}_0=\tilde{\mathcal {R}}_0(\alpha )\ge 0\) exists, and \(\tilde{\mathcal {R}}_0(\alpha )\) is non-increasing in \(\alpha \in {\mathbb {R}}\). Given \(\alpha \in {\mathbb {R}}\), we see from (2.7) that

$$\begin{aligned} \vartheta _t-D\vartheta _{xx}-\alpha \vartheta _x -[g(I_0(t)e^{-k_0x})\vartheta \ge (\lambda _0(\alpha )-\overline{d})\vartheta ,\ \ 0<x<L, 0<t<T, \end{aligned}$$

and

$$\begin{aligned} \vartheta _t-D\vartheta _{xx}-\alpha \vartheta _x -[g(I_0(t)e^{-k_0x})\vartheta \le (\lambda _0(\alpha )-\underline{d})\vartheta ,\ \ 0<x<L, 0<t<T, \end{aligned}$$

where \(\lambda _0(\alpha )=\lambda _0\). With the help of Proposition 5.2, according to our notation, it follows that

$$\begin{aligned} -d_*(\alpha )\ge \lambda _0(\alpha )-\overline{d}, \lambda _0(\alpha )-\underline{d}\ge -d_*(\alpha ), \end{aligned}$$

that is,

$$\begin{aligned} \overline{d}-d_*(\alpha )\ge \lambda _0(\alpha )\ge \underline{d}-d_*(\alpha ). \end{aligned}$$
(5.21)

On the other hand, if \(I_0(t)>0\) for \(t\in [0,T]\) and \(g(I)\ge aI^{\gamma }\) for \(I\in [0,\max _{t\in [0,T]}I_0(t)]\) for some positive constants a and \(\gamma \), it then follows from (Peng and Zhao 2015, Lemma 5.4) that there exists a unique \({\tilde{\alpha }}>0\) such that \(d_\infty (\alpha )>0\) if \(\alpha <{\tilde{\alpha }}\) and \(d_\infty (\alpha )=0\) if \(\alpha \ge {\tilde{\alpha }}\). Recall that \(d_*(\alpha )\) is strictly decreasing in \(L>0\) and in \(\alpha \in {\mathbb {R}}\), \(\lim _{L\rightarrow \infty }d_*(\alpha )=d_\infty (\alpha )\ge 0\), and \(d_\infty (\alpha )\) is a non-increasing function of \(\alpha \in {\mathbb {R}}\). As a result, when \(\alpha \ge {\tilde{\alpha }}\), \(\lambda _0(\alpha )\ge \underline{d}/2>0\) for all large L due to (5.21). We also use \({\mathcal {R}}_0(\alpha )\) instead of \({\mathcal {R}}_0\) to emphasize the dependence of \({\mathcal {R}}_0\) on \(\alpha \). Hence, for \(\alpha \ge {\tilde{\alpha }}\), Lemma 2.2 gives that \(\mathcal {R}_0(\alpha )<1\) for all large L, and so \(\tilde{\mathcal {R}}_0(\alpha )<1\) if \(\alpha \ge {\tilde{\alpha }}\).

We then consider the case of \(\alpha <{\tilde{\alpha }}\). If \(\overline{d}\le d_\infty (\alpha )\), then (5.21), together with the fact that \(d_*(\alpha )\) is strictly decreasing in \(L>0\), implies \(\lambda _0(\alpha )<0\) and so \(\mathcal {R}_0(\alpha )>1\) for all large L due to Lemma 2.2. Consequently, \(\tilde{\mathcal {R}}_0(\alpha )\ge 1\) in this subcase. If \(\underline{d}>d_\infty (\alpha )\), again we can use (5.21) to conclude that \(\lambda _0(\alpha )>0\), which, combined with Lemma 2.2, then shows \(\mathcal {R}_0(\alpha )<1\) for all large L. This also implies \(\tilde{\mathcal {R}}_0(\alpha )<1\) if \(\alpha <{\tilde{\alpha }}\) and \(\underline{d}>d_\infty (\alpha )\) hold.

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Peng, R., Zhao, XQ. A nonlocal and periodic reaction-diffusion-advection model of a single phytoplankton species. J. Math. Biol. 72, 755–791 (2016). https://doi.org/10.1007/s00285-015-0904-1

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