Effects of population and population pressure on forest resources and their conservation: a modeling study

Abstract

In this paper, a nonlinear mathematical model is proposed and analyzed to study the depletion of forest resources caused by population and the corresponding population pressure. It is assumed that the cumulative density of forest resources and the density of populations follow logistic models with prey–predator type nonlinear interaction terms. It is considered that the carrying capacity of forest resources decreases by population pressure, the main focus of this paper. A conservation model is also proposed to control the population pressure by providing some economic incentives to people, the amount of which is assumed to be proportional to the population pressure. The model is analyzed by using stability theory of differential equations and numerical simulation. The model analysis shows that as the density of population or population pressure increases, the cumulative density of forest resources decreases, and the resources may become extinct if the population pressure becomes too large. It is also noted that by controlling the population pressure, using some economic incentives, the density of forest resources can be maintained at an equilibrium level, which is population density dependent. The simulation analysis of the model confirms analytical results.

Appendices

Appendix 1: Proof of theorem 1

Linearizing the model system (1) about E ^*₃ by using the following transformations

$$ B=B^*+b, N=N^*+n, P=P^*+p, $$

(17)

where b, n and p are small perturbations around the equilibrium E ^*₃ . Now using the following positive definite function:

$$ V =\frac{1}{2}\left(\frac{1}{B^*}b^2+\frac{m_1}{N^*}n^2+m_2 p^2\right), $$

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(where m ₁ and m ₂ are some positive constants to be chosen appropriately). Differentiating above equation with respect to t along the solutions of linearized system of (1), we get

$$ \begin{aligned} \frac{{\hbox{d}}V}{{\hbox{d}}t}&=-\left(\frac{s_0}{L}+\lambda_2 P^*\right)b^2 -m_1\frac{r_0}{K}n^2- m_2\lambda_0 p^2\\ &\quad -\left(\alpha-m_1 \pi \alpha \right)b n - \lambda_2 B^* b p + m_2 \lambda p n. \end{aligned} $$

Choosing $m_1= \frac{1}{\pi},\;\frac{{\hbox{d}}V}{{\hbox{d}}t}$ simplified as:

$$ \begin{aligned} \frac{{\hbox{d}}V}{{\hbox{d}}t}&=-\left(\frac{s_0}{L}+\lambda_2 P^*\right)b^2-\lambda_2 B^* b p -\frac{1}{2} m_2 \lambda_0 p^2\\ &\quad -\frac{r_0}{\pi K}n^2 + m_2 \lambda n p - \frac{1}{2}m_2 \lambda_0 p^2. \end{aligned} $$

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We note that $\frac{{\hbox{d}}V}{{\hbox{d}}t}$ is negative definite under condition (8), proving the theorem.

Appendix 2: Proof of theorem 2

To prove this theorem, we consider the following positive definite function.

$$ W = \left(B-B^*-B^* \hbox{ln}\frac{B}{B^*}\right)+k_1\left(N-N^*-N^*\hbox{ln}\frac{N}{N^*}\right) +k_2\frac{(P-P^*)^2}{2}, $$

where k ₁ and k ₂ are positive constants to be chosen appropriately. Now differentiating ‘W’ with respect to ‘t’ along the solutions of model system (1),

$$ \frac{{\hbox{d}}W}{{\hbox{d}}t} = (B-B^*)\frac{\dot{B}}{B}+k_1(N-N^*)\frac{\dot{ N}}{N} +k_2(P-P^*)\dot{P}. $$

Now using model system (1) and making a simple algebraic manipulation, we get

$$ \begin{aligned} \frac{{\hbox{d}}W}{{\hbox{d}}t}&=-(\frac{s_0}{L}+\lambda_2 P)(B-B^*)^2 -\frac{k_1r_0}{K}(N-N^*)^2-\lambda_2 k_2 (P-P^*)^2\\ &\quad - \left(\alpha - \pi \alpha k_1\right)(B - B^*)(N - N^*) - \lambda_2 B^* (B-B^*)(P-P^*)\\ &\quad + k_2 \lambda (N - N^*)(P - P^*). \end{aligned} $$

Now choosing $k_1 = \frac{1}{\pi}$ and using region of attraction, we see that $\frac{{\hbox{d}}W}{{\hbox{d}}t}$ will be negative definite inside the region of attraction provided that condition (9) is satisfied.