Structural changes in Mangroves of Sundarban in Bangladesh: effects of climate change and human disturbances

Abstract

Mangroves refer to the plantation, halophytes growing in the harsh brackish coastal habitats and on riverbanks. These shrubland habitats have adapted to the salinity, low oxygen and wave action with their complex roots and filtration system, despite providing the most coastal protection from natural disasters. This paper explores the response of mangroves of different regions to natural calamities and man-made disasters. A simple Lotka–Volterra model has been developed to see the dynamics of the tidal woods and wetlands. Sundarbans, the largest mangrove of Bangladesh, is victim to a vast amount of hazardous events. Frequencies of cyclones flood and drive the mangroves woody over-story to the ground leaving shrubs and wetland. Some other significant threats are deforestation leading to human settlements for fuel, agriculture, and over-fishing. Whether mangroves can exhibit alternative steady states, while environmental and human influence is considered. This mathematical study of mangroves is unprecedented in such a manner. The conditions for forestland equilibrium to be stable depending on different factors are explored. Small changes in data can result in a vast number of possible outcomes. Data for mangroves of Zambezi River Delta, Mozambique, and Sundarbans are used to generate results with constant human behavior and disasters like floods, cyclones, and tsunamis.

Appendices

Appendix A: Existence and uniqueness

Theorem 1

(Hirsch et al. 2004) Let us consider the initial value problem (IVP),

$$\begin{aligned}&\dot{\underline{x}}=f(\underline{x}), \quad \underline{x}(0)=x_{o}, \end{aligned}$$

(A1)

where $x_{o} \in \mathbb {R}^n$ and $f:\mathbb {R}^n \rightarrow \mathbb {R}^n$ is $C^1$. Then there exist a unique solution of (A1) such that, $p>0$ and it is,

$$\begin{aligned} \underline{x}:(p,p) \rightarrow \mathbb {R}^n \end{aligned}$$

satisfying the initial condition $\underline{x}(0)=x_{o}$.

Proof

Let

$$\begin{aligned}{} & {} f(\underline{x})=(f_{1}(x_{1}, x_{2}, \cdots , x_{n}), \\{} & {} \quad f_{2}(x_{1}, x_{2}, \cdots , x_{n}),\cdots , f_{n}(x_{1}, x_{2}, \cdots , x_{n})). \end{aligned}$$

Let $df_{x}$ be the derivative of f at $\underline{x} \in \underline{x}(0)$, which is a linear map that assigns a unique vector $\underline{v} \in \mathbb {R}^n$ to a point in $\mathbb {R}^n$. In matrix form,

$$\begin{aligned} df_{x} =(\frac{\partial f_{i}}{\partial x_{j}}). \end{aligned}$$

is commonly known as the Jacobian matrix. Since f is $C^1$, which is continuously differentiable and all the partial derivative $f_{j}$ exist and are continuous. Then we can define the norm of $df_{x}$ by

$$\begin{aligned} \Vert df_{x}\Vert =\sup _{\Vert v\Vert =1} \Vert df_{x}(\underline{v})\Vert . \end{aligned}$$

$\square$

Lemma 1

(Hirsch et al. 2004) The function $f:\mathcal {O} \rightarrow \mathbb {R}^n$ is locally Lipschitz if it is $C^1$.

A function $f:\mathcal {O} \rightarrow \mathbb {R}^n$ is known as Lipschitz if for the open set $\mathcal {O} \subset \mathbb {R}^n$, there exists a constant k such that

$$\begin{aligned} \Vert f(y)-f(x)\Vert \leqslant k \Vert y-x\Vert , \quad \forall \;\; x,y \in \mathcal {O}. \end{aligned}$$

Now let us consider an open interval I containing zero and $G:J \rightarrow \mathcal {O}$ that satisfies,

$$\begin{aligned} G'(t)=f(G(t)), \quad G(0)=G_{o}. \end{aligned}$$

(A2)

Integrating equation (A2),

$$\begin{aligned} G(t)=G_{o}+\int _{0}^{t} f(G(s)) \,ds. \end{aligned}$$

(A3)

The equations (A2) and (A3) are equivalent forms for $G:I \rightarrow \mathcal {O}.$ To continue with the proof we present some assumptions. First, $\mathcal {O}_{r}$ is a closed ball of radius $r>0$ centered at $x_{o}$. There exist a Lipschitz constant k for f on $\mathcal {O}_{r}$ and $f(\underline{x})<M$. Lastly, we choose $p< \min \{\frac{r}{M},\frac{1}{k}\}$ and let $I=[-p,p].$

Lemma 2

(Hirsch et al. 2004) For a sequence of continuous functions $\phi _{j}:I \rightarrow \mathbb {R}^n,\; j=0,1,2,\cdots ,n$ defined on a closed interval I and given, $\epsilon >0$ and some $N>0$, for every $m,n>N$,

$$\begin{aligned} \max _{t \in I} \Vert \phi _{m}(t)-\phi _{n}(t)\Vert < \epsilon . \end{aligned}$$

Then there is a continuous function $\phi :I \rightarrow \mathbb {R}^n$ such that

$$\begin{aligned} \max _{t \in I} \Vert \phi _{j}(t)-\phi (t)\Vert < 0, \quad \text {as } k \rightarrow \infty . \end{aligned}$$

Now from (A3),

$$\begin{aligned}&\phi _{j+1}(t)=G_{o}+\int _{0}^{t} f(\phi _{j}(s)) \,ds \nonumber \\&\Rightarrow \lim _{j \rightarrow \infty }\phi _{j+1}(t)=G_{o}+\lim _{j \rightarrow \infty }\int _{0}^{t} f(\phi _{j}(s)) \,ds \nonumber \\&\Rightarrow G(t)=G_{o}+\int _{0}^{t} (\lim _{j \rightarrow \infty }f(\phi _{j}(s))) \,ds =G_{o}+\int _{0}^{t} f(G(s)) \,ds. \end{aligned}$$

(A4)

Hence, $G:I \rightarrow \mathcal {O}_{r}$ satisfies the integral form of the differential equation and therefore is the solution of the equation. Now, we pay attention to the uniqueness part of the theorem. Let us consider the two solutions $P,Q:I\rightarrow \mathcal {O}_{r}$ of the differential equation (A1) that satisfy $P(0)=Q(0)=G_{0}$, where the interval I is same as above. Let,

$$\begin{aligned} M= \max _{t \in I} \Vert P(t)-Q(t)\Vert , \end{aligned}$$

where the maximum value is attained at some arbitrary point $t_{i} \in I$. Therefore,

$$\begin{aligned}{} & {} \Vert P(t)-Q(t)\Vert = \Vert \int _{0}^{t_{i}} P'(s)-Q'(s)\,ds\Vert \nonumber \\{} & {} \quad \leqslant \int _{0}^{t_{i}} \Vert f(P(s))-f(Q(s))\Vert \,ds \leqslant \nonumber \\{} & {} \quad \int _{0}^{t_{i}} k\Vert P(s)-Q(s)\Vert \,ds \leqslant pkM. \end{aligned}$$

(A5)

Now for the inequality (A5) to hold M must be 0, since $pk<1$. Thus, we have

$$\begin{aligned} P(t) \equiv Q(t). \end{aligned}$$

Appendix B: Sensitivity analysis

In this section, we computed the sensitivities of the dimensionless parameters of equation (4). In (Carboni et al. 2007), various types of sensitivities such as local sensitivity, square normalized local sensitivities on multiple biological models were discussed. Let us consider the system,

$$\begin{aligned} {\left\{ \begin{array}{ll} \dot{x_{1}}=&{}f_{1}(x_{1},x_{2},\dots ,x_{n}) \\ \dot{x_{2}}=&{}f_{2}(x_{1},x_{2},\dots ,x_{n}) \\ \vdots &{} \vdots \\ \dot{x_{m}}=&{}f_{m}(x_{1},x_{2},\dots ,x_{n}) \\ \end{array}\right. } \end{aligned}$$

(B6)

In vector form, the system (B6), $\dot{\underline{x}}=f(\underline{x})$. We used local forward sensitivity analysis here for ordinary differential equations.

$$\begin{aligned}&\frac{\displaystyle d \underline{u}}{\displaystyle d p_{j}}=\frac{\displaystyle \partial f}{\displaystyle \partial \underline{x}} \underline{u}+\frac{\displaystyle \partial f}{\displaystyle \partial p_{j}} \nonumber \\&\quad \Rightarrow \frac{\displaystyle d \underline{u}}{\displaystyle d p_{j}}&=J.\underline{u}+F_{j}. \end{aligned}$$

(B7)

Here, in equation (B7), $J=\frac{\displaystyle \partial f}{\displaystyle \partial \underline{x}}$ is the Jacobian of the system (B6). $F_{j}=\frac{\displaystyle \partial f}{\displaystyle \partial p_{j}}$ is the partial derivatives of the equations of (B6) with respect to respected parameters $p_{j}$ of the system, and $\frac{\displaystyle d u}{\displaystyle d p_{j}}$ is corresponding vector of sensitivities. The vector $\underline{u}$ will have the same number of elements as the number of equations.

Here in Fig. 9, we have solved the systems obtained through (B7) for the parameters $a, \lambda _{h}, \lambda _{u}, \text {and } \lambda _{w}$, respectively. The Jacobian of system (4) is,

$$\begin{aligned} J= \begin{bmatrix} -\lambda _{h}-\lambda _{u}-a*p-a*(p-1) &{} 0 \\ -s &{} 1-2s-p--\lambda _{w}. \end{bmatrix} \end{aligned}$$

(B8)

First, we consider the parameter a and let $j=1$.The matrix $F_{1}$ for the first parameter a is

$$\begin{aligned} F_{1}= \begin{bmatrix} -p(p-1) \\ 0 \end{bmatrix} \end{aligned}$$

(B9)

Substituting the values from Eqs. (B8) and (B9) into (B7), we obtain

$$\begin{aligned}{} & {} \begin{bmatrix} \frac{\displaystyle d u_{1}}{\displaystyle d t} \\ \frac{\displaystyle d u_{2}}{\displaystyle d t} \end{bmatrix} = \begin{bmatrix} -\lambda _{h}-\lambda _{u}-a*p-a*(p-1) &{} 0 \\ -s &{} 1-2s-p--\lambda _{w} \end{bmatrix}\\{} & {} \quad \begin{bmatrix} u_{1} \\ u_{2} \end{bmatrix} + \begin{bmatrix} -p(p-1) \\ 0 \end{bmatrix} \end{aligned}$$

$$\begin{aligned}{} & {} \Rightarrow \begin{bmatrix} \frac{\displaystyle d u_{1}}{\displaystyle \textrm{d} t} \\ \frac{\displaystyle d u_{2}}{\displaystyle \textrm{d} t} \end{bmatrix} =\nonumber \\{} & {} \quad \begin{bmatrix} -p(p-1) -u_{1}(\lambda _{h}+\lambda _{u}+a*p+a*(p-1))\\ -s u_{1}-u_{2}(\lambda _{w}+p+2s-1) \end{bmatrix} \end{aligned}$$

(B10)

Solving the system (B10), we obtain Fig. 9a for parameter a. In similar manner we can do sensitivity analysis for rest parameters.

Fig. 9

Forward sensitivity analysis for dimensionless parameters $a, \lambda _{h}, \lambda _{u}, \lambda _{w}$ of Eq. (4). The parameter values are provided in Table 7

Table 7 Description and estimations of parameters for sensitivity analysis.