## 2 The Models and Analyses  
The model in [PERSON] and [PERSON] [25] describing the habitat proportion occupied by species \(i\), \(p_{i}\), can be written as follows:  
\[ \begin{array}{c}F_{1}=\frac{dp_{1}}{dt}=c_{1}p_{1}(h(t)-p_{1})-m_{1}p_{1}+f(t )p_{1}\\ F_{i}=\frac{dp_{i}}{dt}=c_{i}p_{i}\Bigg{(}h(t)-\sum\limits_{j=1}^{i}p_{j} \Bigg{)}-m_{i}p_{i}-\sum\limits_{j=1}^{i-1}c_{j}p_{i}p_{j}+f(t)p_{i},i=2,3, \ldots,n\end{array}  \tag{1}\]  
where \(c_{i}\) is the colonization or invasion rate of \(ith\) species. In this study, the \((i+1)^{th}\) species is a better colonization or invasion species compared to the \(ith\) species,  
\(m_{i}\) is the mortality rate of \(ith\) species,  
\(h(t)\) represents the remaining habitat,  
\(f(t)p_{i}\) is the direct effect of habitat destruction where \(f(t)=\frac{h^{\prime}(t)}{h(t)}\) is the ratio between the rate of change of the remaining habitat and the remaining habitat; in other words, \(f(t)\) is the balance between the two figures. If \(f(t)\) is a decreasing function, the direct effect of the habitat loss will be decreasing. Otherwise, it will be increasing over time. The Jacobian of the system (1) can be expressed as a lower triangular matrix:  
\[J= \begin{bmatrix}c_{1}h-2c_{1}p_{1}-m_{1}+f&0&\cdots&\cdots&0\\ -(c_{2}+c_{1})p_{2}&c_{2}h-(c_{2}+c_{1})p_{1}-2c_{2}p_{2}-m_{2}+f&0&\cdots&0\\ -(c_{3}+c_{1})p_{3}&-(c_{3}+c_{2})p_{3}&\frac{\partial F_{1}}{\partial p_{3}} &\cdots&0\\ \vdots&\vdots&\cdots&\cdots&\vdots\\ -(c_{n}+c_{1})p_{n}&-(c_{n}+c_{2})p_{n}&\cdots&-c_{n}p_{n}-c_{n-1}p_{n}&\frac{ \partial F_{n}}{\partial p_{n}}\end{bmatrix} \]  
where \(\frac{\partial F_{j}}{\partial p_{j}}\), the \(jth\) diagonal element is \(c_{j}h-c_{j}\sum\limits_{i=1}^{j-1}p_{i}-\sum\limits_{i=1}^{j-1}c_{i}p_{i}-2 c_{j}p_{j}-m_{j}+f\). System (1) has the trivial equilibrium point at \(p_{i}=0,\forall i\) with the Jacobian at the equilibrium:  
\[J_{0}= \begin{bmatrix}c_{1}h-m_{1}+f&0&\cdots&0\\ 0&c_{2}h-m_{2}+f&&\vdots\\ \vdots&&\ddots&0\\ 0&\cdots&0&c_{n}h-m_{n}+f\end{bmatrix} \]  
From the structure of \(J_{o}\), the linearized system is decoupled. Therefore, to guarantee the survival of the \(ith\) species, it is sufficient to show that there exists a time \(T\geq 0\) in which the \(ith\) term of the diagonal of \(J_{o}\) (or the \(ith\) eigenvalue of \(J_{o}\)) is positive for all \(t\geq T\). However, it does not mean that the species having negative eigenvalues for all \(t\geq 0\) will be extinct since the associated solution of (1) might not converge to the origin depending on the initial condition. There are several special cases worth exploiting explicitly for the trivial equilibrium case:  
Case 1: \(c_{i}h-m_{i}+f(t)=\lambda_{i}\) case when \(\lambda_{i}\) are constant and \(h(t)\) is a solution to the equation:  
\[h^{\prime}+k_{i}h=c_{i}h^{2}\]  
where \(k_{i}=\lambda_{i}+m_{i}\) are constant. Solving the above differential equation yields  
\[h=k_{i}/\left(c_{i}+\partial k_{i}e^{-k_{i}t}\right),\]  
where \(\partial=\frac{1}{h(0)}-\frac{c_{i}}{k_{i}}\). Hence, in order for \(h=\frac{k_{i}}{c_{i}+\partial k_{i}e^{-k_{i}t}},\forall i\), we must have \(k_{i}=k\) and \(c_{i}=c\) for some constants \(k\) and \(c\). Thus, in this case the habitat is equally distributed among species and \(\lambda_{i}=k-m_{i}\). Hence, for any \(i\), if \(\lambda_{i}>0\), the \(ith\) species survives the habitat destruction. **Remark 1**. : _For constant \(h\), i.e., \(f(t)=0\), we have that \(\lambda_{i}=c_{i}h-m_{i}\) is constant and is a special case to this case._  
Case 2: A more general \(c_{i}h-m_{i}+f(t)\) case. In order for the \(ith\) species to survive the habitat loss, \(c_{i}h-m_{i}+f(t)\) needs to be positive, \(\forall t>T\), for some large enough T.  
**Remark 2**. : _We assume in this study that \(c_{i+1}>c_{i}\), \(\forall i\in\{1,2,\ldots,n\}\).