Mathematical model to study the impact of anthropogenic activities on forest biomass and forest-dependent wildlife population

Abstract

This paper proposes and analyses a nonlinear mathematical model to study the impact of anthropogenic activities on forest biomass and forest-dependent wildlife populations using a system of differential equations. It is assumed that the growth of forest biomass, forest-dependent wildlife populations, and the human population follow logistic equations. The effect of forest biomass depletion on the survival of forest-dependent wildlife populations is investigated by introducing a function that denotes the dependence on forest biomass. The system’s behaviour near all ecologically acceptable equilibria is studied, and to confirm the analytical conclusions, a numerical simulation is performed. The model analysis shows that as forest biomass declines due to an increase in human population and its associated activities, the population of wildlife species also declines, and if no measures are taken, both forest biomass and the wildlife population may become extinct.

Appendices

Appendix A: Proof of Lemma 1

Proof

Following Chaudhary et al. [28] and Chen [29], the proof is as follows. Consider the first equation of the system (1). When $B \rightarrow L$ the rest of the variables become zero, this gives,

$$\begin{aligned} \dfrac{{\text {d}}B}{{\text {d}}t}\le sB \left( 1-\dfrac{B}{L}\right) \implies \limsup _{t\rightarrow \infty } B\le L. \end{aligned}$$

Further, from equations two, three, and four of the system, respectively, we get Eqs. (A1), (A2) and (A3).

$$\begin{aligned}{} & {} \dfrac{{{\text {d}}W}}{{{\text {d}}t}}\le {r(L)W}\left( 1-\dfrac{{W}}{{K(L)}}\right) \implies \limsup _{t\rightarrow \infty } {W}\le {K(L)}. \nonumber \\ \end{aligned}$$

(A1)

$$\begin{aligned}{} & {} \dfrac{{{\text {d}}N}}{{{\text {d}}t}}\le {\theta N}\left( 1 -\dfrac{{N}}{{M}}\right) + \lambda \beta _1LN,\nonumber \\{} & {} \qquad \implies \limsup _{{t}\rightarrow \infty }{N}\le \dfrac{{M}}{{\theta }}\left( \theta +\lambda \beta _1 L \right) ={N_{m}} \quad \mathrm{(Say)}. \nonumber \\\end{aligned}$$

(A2)

$$\begin{aligned}{} & {} \dfrac{{{\text {d}}H}}{{{\text {d}}t}}\le \gamma N_m-\gamma _1H,\nonumber \\{} & {} \qquad \implies \limsup _{t\rightarrow \infty }{H}\le \dfrac{{\gamma N_m}}{\gamma _1}={H_{m} }\quad \mathrm{(Say)}. \end{aligned}$$

(A3)

$\square$

Appendix B: Proof of Theorem 3

Proof

To prove Theorem (3), we start by linearising the system (1), about $E^*$ by using the following transformations:

$B=B^*+b$, $W=W^*+w$, $N=N^*+n$, $H=H^*+h$, where b, w, n and h are small perturbations around the equilibrium $E^*$. The following linearised system is obtained:

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{{\text {d}}b}{{\text {d}}t}{=}{-}\left( \dfrac{sB^*}{L}{+}\beta _2B^*H^* \right) b{-}\alpha B^*w{-}\beta _1B^*n{-}\beta _2{B^{*}}^2h ,\\ \dfrac{{\text {d}}w}{{\text {d}}t}{=} \left( r'(B^*) W^*\left( 1{-}\dfrac{W^*}{K(B^*)} \right) {+}\dfrac{r(B^*) {W^*}^2K'(B^*)}{{K(B^*)}^*2} \right) b \\ ~~~~~~~~-\left( \dfrac{r(B^*) W^*}{K(B^*)} \right) w-\nu _1W^*n{-}\nu _2W^*h,\\ \dfrac{{\text {d}}n}{{\text {d}}t}=\lambda \beta _1N^*b-\sigma N^*w-\dfrac{\theta N^*}{M}n,\\ \dfrac{{\text {d}}h}{{\text {d}}t}=\gamma n -\gamma _1h. \end{array}\right. } \nonumber \\ \end{aligned}$$

(B4)

Following Lata [10], we considered the positive definite function (B5),

$$\begin{aligned} V=\dfrac{1}{2}\left( \dfrac{1}{B^*}b^2+\dfrac{k_1}{W^*}w^2+k_2n^2+k_3h^2 \right) , \end{aligned}$$

(B5)

where $k_1$, $k_2$ and $k_3$ are positive constants. Upon differentiating function (B5) with respect to t along the solutions of linearized system (B4), the following is obtained:

$$\begin{aligned} \dfrac{{\text {d}}V}{{\text {d}}t}={} & {} -\left( \dfrac{s}{L}+\beta _2H^* \right) b^2- \alpha bw-\beta _1bn-\beta _2 B^*bh\nonumber \\{} & {} -k_1\left( \dfrac{r(B^*)}{K(B^*)} \right) w^2-k_1\nu _1wn \nonumber \\{} & {} -k_1\nu _2wh +k_1\left( r'(B^*)\left( 1-\dfrac{W^*}{K(B^*)} \right) \right. \nonumber \\{} & {} \left. +\dfrac{r(B^*) {W^*}K'(B^*)}{{K(B^*)}^2} \right) bw \nonumber \\{} & {} +k_2\lambda \beta _1N^*bn-k_2N^*\sigma wn-k_2\dfrac{\theta N^*}{M}n^2 \nonumber \\{} & {} +k_3\gamma nh - k_3\gamma _1 h^2. \end{aligned}$$

(B6)

Eq. (B6) can be arranged as follows:

$$\begin{aligned} \dfrac{{\text {d}}V}{{\text {d}}t}={} & {} -\left( \dfrac{s}{L}+\beta _2H^* \right) b^2-k_1\left( \dfrac{r(B^*)}{K(B^*)} \right) w^2\nonumber \\{} & {} -k_2\dfrac{\theta N^*}{M}n^2-k_3 \gamma _1 h^2\nonumber \\{} & {} +\left( -\alpha +k_1r'(B^*) \right) bw-k_1\left( \dfrac{r'(B^*)W^*}{K(B^*)}\right. \nonumber \\{} & {} \left. -\dfrac{r(B^*) {W^*}K'(B^*)}{{K(B^*)}^2} \right) bw+\left( -\beta _1+k_2\lambda \beta _1 N^* \right) bn\nonumber \\{} & {} -\beta _2 bh -\left( k_1\nu _1+k_2\sigma N^*\right) wn\nonumber \\{} & {} -k_1\nu _2wh+k_3\gamma nh. \end{aligned}$$

(B7)

Choosing $k_1=\dfrac{1}{\eta },$ $k_2=\dfrac{1}{\lambda N^*},$ and $k_3=1$ which are all positive constants, Eq. (B7) is written as the sum of quadratic expressions

$$\begin{aligned} \dfrac{{\text {d}}v}{{\text {d}}t}={} & {} -\left[ \dfrac{1}{2}\left( \dfrac{s}{L}+\beta _2H^* \right) b^2+k_1\left( \dfrac{r'(B^*)W^*}{K(B^*)}\right. \right. \nonumber \\{} & {} \left. \left. -\dfrac{r(B^*)K'(B^*)W^*}{(K(B^*))^2}\right) bw+\dfrac{k_1}{3}\left( \dfrac{r(B^*)}{K(B^*)} \right) w^2 \right] \nonumber \\{} & {} -\left[ \dfrac{k_1}{3}\left( \dfrac{r(B^*)}{K(B^*)}\right) w^2+\left( k_1\nu _1+k_2\sigma N^* \right) wn\right. \nonumber \\{} & {} \left. +\dfrac{k_2}{2}\left( \dfrac{\theta N^*}{M}\right) n^2\right] \nonumber \\{} & {} -\left[ \dfrac{k_2}{2}\left( \dfrac{\theta N^*}{M} \right) n^2-k_3\gamma nh +\dfrac{k_3}{3} \gamma _1h^2 \right] \nonumber \\{} & {} -\left[ \dfrac{k_3}{3} \gamma _1 h^2+k_1\nu _2 wh+\dfrac{k_1}{3}\left( \dfrac{r(B^*)}{K(B^*)}\right) w^2 \right] \nonumber \\{} & {} -\left[ \dfrac{1}{2}\left( \dfrac{s}{L}+\beta _2H^* \right) b^2 +\beta _2bh+ \dfrac{k_3}{3} \gamma _1 h^2\right] \end{aligned}$$

(B8)

When $\dfrac{{\text {d}}V}{{\text {d}}t}<0$, the model system (1) becomes locally asymptotically stable in the given region of attraction. Therefore, $\dfrac{{\text {d}}V}{{\text {d}}t}$ is negative definite if and only if conditions (18)–(22) hold. $\square$

Appendix C: Proof of Theorem 4

Proof

We consider the following positive definite function [8] to prove the theorem

$$\begin{aligned} U{=}{} & {} \left( B{-}B^*{-}B^*\ln {\dfrac{B}{B^*}}\right) {+}l_1\left( W{-}W^*{-}W^*\ln {\dfrac{W}{W^*}} \right) \nonumber \\{} & {} +l_2\left( N-N^*-N^*\ln {\dfrac{N}{N^*}} \right) \nonumber \\{} & {} +\dfrac{l_3}{2}\left( H-H^* \right) ^2, \end{aligned}$$

(C9)

where $l_1$, $l_2$ and $l_3$ are positive constants. It can be easily checked that, $U(B^*, W^*, N^*, H^*)=0$ and $U(B, W, N, H)>0$ for all other positive values of $\left\{ B, W, N, H\right\} -\left\{ B^*, W^*, N^*,\right.$$\left. H^* \right\}$.

On differentiating (C9) with respect to "t", the following is obtained:

$$\begin{aligned} \dfrac{{\text {d}}U}{{\text {d}}t}= & {} \left( \dfrac{B-B^*}{B}\right) \dfrac{{\text {d}}B}{{\text {d}}t}+l_1\left( \dfrac{W-W^*}{W}\right) \dfrac{{\text {d}}W}{{\text {d}}t}\\{} & {} +l_2\left( \dfrac{N-N^*}{N}\right) \dfrac{{\text {d}}N}{{\text {d}}t}+l_3\left( H-H^*\right) \dfrac{{\text {d}}H}{{\text {d}}t}. \end{aligned}$$

Substituting the values of $\dfrac{{\text {d}}B}{{\text {d}}t}$, $\dfrac{{\text {d}}W}{{\text {d}}t}$, $\dfrac{{\text {d}}N}{{\text {d}}t}$ and $\dfrac{{\text {d}}H}{{\text {d}}t}$ from the system (1), the following equation is obtained;

$$\begin{aligned} \dfrac{{\text {d}}U}{{\text {d}}t}{=}{} & {} \left( B{-}B^* \right) \left[ -\left( \dfrac{s}{L}{+}\beta _2H^*\right) \left( B-B^*\right) -\alpha (W-W^*)\right. \\{} & {} \left. -\beta _1(N-N^*)-\beta _2 B (H-H^*) \right] \\{} & {} +l_1\left( W-W^* \right) \left[ \alpha \eta \left( B-B^*\right) -\dfrac{\alpha \eta B}{K(B^*)}(W-W^*)\right. \\{} & {} \left. -\nu _1 (N-N^*)-\nu _2 (H-H^*)\right] \\{} & {} +l_1\left( W-W^* \right) \left[ -\left( \alpha \eta B W \varGamma (B)\right) \left( B-B^*\right) \right. \\{} & {} \left. -\left( \dfrac{\alpha \eta W^*}{K(B^*)}\right) \left( B-B^*\right) \right] \\{} & {} +l_2(N-N^*)\left[ -\dfrac{\theta }{M}(N-N^*)+\lambda \beta _1 (B-B^*)\right. \\{} & {} \left. -\sigma (W-W^*) \right] \\{} & {} +l_3(H-H^*)\left[ -(\gamma _1)(H-H^*)+\gamma (N-N^*)\right] . \end{aligned}$$

where,

$$\begin{aligned} \varGamma (B)= \left\{ \begin{array}{ll} \dfrac{\dfrac{1}{K(B)}-\dfrac{1}{K(B^*)}}{B-B^*}, &{}\quad B\ne B^*\\ \dfrac{-K'(B^*)}{(K(B^*))^2}, &{}\quad B=B^*. \end{array} \right. \end{aligned}$$

Utilizing mean value theorem, we have

$$\begin{aligned} \left| {\varGamma (B)}\right| = \dfrac{\lambda _1}{K_0^2}, \end{aligned}$$

such that $0<K'(B)\le \lambda _1.$ Choosing $l_1=\dfrac{1}{\eta }$, $l_2=\dfrac{1}{\lambda }$ and $l_3= 1$, consequently, $\dfrac{{\text {d}}U}{{\text {d}}t}$ is reduced to:

$$\begin{aligned} \dfrac{{\text {d}}U}{{\text {d}}t} {=}{} & {} {-}\left[ \dfrac{1}{2}\left( \dfrac{s}{L}{+}\beta _2H^*\right) \left( B{-}B^*\right) ^2{+}l_1\zeta (B{-}B^*)(W{-}W^*)\right. \nonumber \\{} & {} \left. +\dfrac{l_1}{3}\dfrac{\alpha \eta B}{K(B^*)}\left( W-W^*\right) ^2 \right] \nonumber \\{} & {} -\left[ \dfrac{l_1}{3}\dfrac{\alpha \eta B}{K(B^*)}\left( W-W^*\right) ^2+\left( l_1\nu _1+l_2\sigma \right) (W-W^*)\right. \nonumber \\{} & {} \left. (N-N^*)+\dfrac{l_2\theta }{2M}(N-N^*)^2\right] \nonumber \\{} & {} -\left[ \dfrac{l_2\theta }{2M}(N-N^*)^2-l_3\gamma (N-N^*)(H-H^*)\right. \nonumber \\{} & {} \left. +\dfrac{l_3}{3}\gamma _1\left( H-H^*\right) ^2 \right] \nonumber \\{} & {} -\left[ \dfrac{l_3}{3}\gamma _1\left( H-H^*\right) ^2 +\beta _2 B(B-B^*)(H-H^*)\right. \nonumber \\{} & {} \left. +\dfrac{1}{2}\left( \dfrac{s}{L}+\beta _2H^*\right) \left( B-B^*\right) ^2\right] \nonumber \\{} & {} {-}\left[ \dfrac{l_1}{3}\dfrac{\alpha \eta B}{K(B^*)}\left( W{-}W^*\right) ^2{+}l_1\nu _2(W-W^*)(H{-}H^*)\right. \nonumber \\{} & {} \left. +\dfrac{l_3}{3} \gamma _1\left( H-H^*\right) ^2 \right] . \end{aligned}$$

(C10)

Where,

$$\begin{aligned} \zeta =\left[ \left( \alpha \eta B W \varGamma (B)\right) +\left( \dfrac{\alpha \eta W^*}{K(B^*)}\right) \right] . \end{aligned}$$

When $\dfrac{{\text {d}}U}{{\text {d}}t}<0$, the model system (1) becomes globally asymptotically stable in the region of attraction $E^*$. Therefore, $\dfrac{{\text {d}}U}{{\text {d}}t}$ is negative if and only if conditions (23)–(27) hold. Theorem 4 follows. $\square$

Proof

Following Chen [29], we begin by defining the following condition: The model system (1) is said to be uniformly persistent if there are positive constants $\varOmega _1$ and $\varOmega _2$ such that each of the positive solutions $\left( B, W, N, H \right)$ of the system with positive initial conditions $\left( B_0, W_0, N_0, H_0 \right)$ satisfies Eq. (C11).

$$\begin{aligned} \varOmega _1\le \liminf _{t\rightarrow \infty } \mathcal {X}(t)\le \limsup _{t\rightarrow \infty }\mathcal {X}(t)\le \varOmega _2 \end{aligned}$$

(C11)

where $\mathcal {X}(t)=\left( B, W, N, H \right)$.

From Lemma (1), we define $\varOmega _2$ as

$$\begin{aligned} \varOmega _2=\max \left\{ L, K(L), N_m, H_m\right\} . \end{aligned}$$

It follows that $\limsup _{t\rightarrow \infty }\mathcal {X}(t)\le \varOmega _2.$

This is also true for any sufficiently small $\epsilon _i>0$, there exists a $T_i>0$, such that

$$\begin{aligned}{} & {} B<L+\epsilon _1 \quad \forall t\ge T_1,\\{} & {} W<K(L)+\epsilon _2 \quad \forall t\ge T_2,\\{} & {} N<N_m+\epsilon _3 \quad \forall t\ge T_3,\\{} & {} H<H_m+\epsilon _4 \quad \forall t\ge T_4. \end{aligned}$$

Now, taking $\epsilon =\max \left\{ \epsilon _1, \epsilon _2, \epsilon _3, \epsilon _4\right\}$ and $T=\max \left\{ T_1,\right.$$\left. T_2, T_3, T_4 \right\}$ it can be concluded that, for $\epsilon >0$, there exists $T>0$ such that for all $t\ge T$, the following holds;

$$\begin{aligned} B<L+\epsilon ,\quad W<K(L)+\epsilon , \quad N<N_m+\epsilon , \quad H<H_m+\epsilon . \end{aligned}$$

Thus, the first equation of the model system (1), for all $t\ge T$, can be written as

$$\begin{aligned} \begin{aligned} \dfrac{{\text {d}}B}{{\text {d}}t}&\ge sB \left( 1-\dfrac{B}{L}\right) -\alpha B(K(L)+\epsilon )\\&\quad -\beta _{1}B(N_m+\epsilon )-\beta _{2}B^2(H_m+\epsilon ),\\&=\left( s-\alpha (K(L)+\epsilon )-\beta _1(N_m+\epsilon ) \right) B\\&\quad -\left( \dfrac{s}{L}+\beta _2(H_m+\epsilon ) \right) B^2. \end{aligned} \end{aligned}$$

Further, we have

$$\begin{aligned} \liminf _{t\rightarrow \infty }B(t)\ge \dfrac{L(s-\alpha (K(L)+\epsilon )-\beta _1(N_m+\epsilon ))}{s+L\beta _2(H_m+\epsilon )}, \end{aligned}$$

which is true for every $\epsilon >0$, hence,

$$\begin{aligned} \liminf _{t\rightarrow \infty }B(t)\ge \dfrac{L(s-\alpha K(L)-\beta _1N_m)}{s+L\beta _2H_m}= B_l \quad \mathrm{(Say)}, \end{aligned}$$

such that $s>\alpha K(L)+\beta _1N_m.$

From the second model equation, we have

$$\begin{aligned} \begin{aligned} \dfrac{{\text {d}}W}{{\text {d}}t}&\ge r(B_l)W\left( 1-\dfrac{W}{K(B_l)}\right) -\nu _{1}W(N_m+\epsilon )\\&\quad -\nu _{2}W(H_m+\epsilon ),\\&=\left( r(B_l)-\left( \nu _1(N_m+\epsilon )+\nu _2(H_m+\epsilon ) \right) \right) W \\&\quad -\dfrac{r(B_l)}{K(B_l)}. \end{aligned} \end{aligned}$$

It follows

$$\begin{aligned} \liminf _{t\rightarrow \infty }W(t)\ge \dfrac{K(B_l)(r(B_l){-}\nu _1(N_m{+}\epsilon ){-}\nu _2(H_m{+}\epsilon ))}{r(B_l)}, \end{aligned}$$

which is true for every $\epsilon >0$, therefore,

$$\begin{aligned} \liminf _{t\rightarrow \infty }W(t)\ge \dfrac{K(B_l)(r(B_l){-}\nu _1N_m{-}\nu _2H_m)}{r(B_l)} {=} W_l \quad \mathrm{(Say)}, \end{aligned}$$

where $r(B_l)>\nu _1N_m+\nu _2H_m$.

Similarly, the third and fourth equations of the system (1), respectively, yield

$$\begin{aligned} \begin{aligned} \dfrac{{\text {d}}N}{{\text {d}}t}&\ge \theta N\left( 1-\dfrac{N}{M}\right) -\sigma NW_l,\\&=\left( \theta -\sigma W_l \right) N -\dfrac{\theta N^2}{M}. \end{aligned} \end{aligned}$$

Hence, it follows

$$\begin{aligned} \liminf _{t\rightarrow \infty }N(t)\ge \dfrac{M(\theta -\sigma W_l)}{\theta }= N_l \quad \mathrm{(Say)}, \end{aligned}$$

where $\theta >\sigma W_l.$

$$\begin{aligned} \begin{aligned} \dfrac{{\text {d}}H}{{\text {d}}t}&\ge \gamma N_l-\gamma _{1}H,\\&=\gamma N_l-\gamma _1H. \end{aligned} \end{aligned}$$

Therefore, it follows

$$\begin{aligned} \liminf _{t\rightarrow \infty }H(t)\ge \dfrac{\gamma N_l}{\gamma _1}=H_l \quad \mathrm{(Say)}. \end{aligned}$$

It is worth noting that $\varOmega _1=\min \left\{ B_l, W_l, N_l, H_l \right\} .$ Theorem 5 follows. $\square$