Immediate and cumulative stresses associated with the multiscale impacts of ecotourism on ecological status and resilience

Abstract

Anthropogenic impacts on ecosystems across spatiotemporal scales are expanding globally, undermining ecosystem resilience and increasing the risk of regime shifts within ecosystems. Governance incorporating social–ecological considerations has become essential. Here, we investigate two kinds of spatiotemporal multiscale impacts of ecotourism: “fast-acting and spatially-limited” impacts, and “slow-acting and large-scale” impacts. We showed that high levels of ecotourism impacts operating at multiple scales can generate an alternative stable state, and a potential simplification of our discussion resorting to the existence of multiple scales in the system. Our study provides insights into resilient ecotourism under the multiscale dynamics that can inform management decisions at appropriate scales.

Appendices

Appendix

Immediate and cumulative stresses associated with the multiscale impacts of ecotourism on ecological status and resilience (Fig. 5, 6)

Fig. 5

Stability diagram of the social–ecological system, overlapping with scaled population size ($x^*/K$) at a state of equilibrium (a-d for $q=2$ and i-l for $q=20$) and equilibrium values of the cumulative impact ($z^*$; e-h for $q=2$ and m-p for $q=20$). Other parameter values used were $\alpha _2=0.8$, $\alpha _4=0.1$

Fig. 6

Stability diagram of the social–ecological system, overlapping with scaled population size ($x^*/K$) at a state of equilibrium (a-d for $q=2$ and i-l for $q=20$) and equilibrium values of the cumulative impact ($z^*$; e-h for $q=2$ and m-p for $q=20$). Other parameter values used were $\alpha _2=1.2$, $\alpha _4=0.1$

A Mathematical details

A.1 The social–ecological model

Considering an ecological status of x (e.g., forest cover or the number of a concerned species), with y number of ecotourists and the degree of cumulative impact z (e.g., habitat degradation or pollution levels) the system dynamics can be expressed as follows:

$$\begin{aligned}&\frac{\mathrm{d}x}{\mathrm{d}t} = \underbrace{r_x x \left( 1-\frac{x}{K} \right) }_\mathrm{Logistic~growth} - \underbrace{\frac{a_y xy}{A_y+x}}_\mathrm{Immediate~impact} - \underbrace{a_z xz}_{\mathrm{Cumulative~impact}}, \end{aligned}$$

(A.1a)

$$\begin{aligned}&\frac{\mathrm{d}y}{\mathrm{d}t} = \underbrace{\frac{a_x x^q}{A_x^q+x^q}}_\mathrm{Ecotourism~attraction} -\underbrace{d_y y}_{\mathrm{Ecotourists~exit}}, \end{aligned}$$

(A.1b)

$$\begin{aligned}&\frac{\mathrm{d}z}{\mathrm{d}t} = \underbrace{r_y y}_{\mathrm{Production}} - \underbrace{d_z z}_{\mathrm{Decomposition}}, \end{aligned}$$

(A.1c)

where $r_x$ and K, respectively, denote the growth rate and carrying capacity of ecological status. The second and the third terms in Eq. (A.1a) denote the immediate impact of ecotourism (with the coefficient $a_y$ and half-saturation point $A_y$) and its cumulative impact (with coefficient $a_z$) on ecological status.

The first term in Eq. (1b) denotes the rate of increase of ecotourists, with $a_x$ as the maximum rate of ecotourist growth, $d_y$ as the ecotourist exit rate, and q as the exponent that determines the shape of the function. As ecotourists are short-term visitors, $a_x$ and $d_y$ must be of the same order of magnitude. The cumulative impact depends on the growth/emission rate of a pollutant caused by ecotourists, $r_y$, of ecotourists and is reduced at the degradation rate, $d_z$.

The number of equilibrium points in Eq. (A.1)

Instead of numerically calculating the number of equilibrium states from the full model Eq. (A.1), we counted the number of points that crosses the value $\mathrm{d}x/\mathrm{d}t=0$ in Eq. (A.1a) after substituting equilibrium states of y and z. This approach enabled us to reduce the number of parameters by scaling with appropriate parameters. These results were also used to derive the bifurcation diagram.

We set $\mathrm{d}y/\mathrm{d}t=0$ in Eq. (A.1b) and solved for y. We also set $\mathrm{d}z/\mathrm{d}t=0$ in Eq. (A.1c) and solved for z. These results were then inserted into Eq. (A.1a) using the setting $\mathrm{d}x/\mathrm{d}t=0$ to obtain the following equation:

$$\begin{aligned} x \left\{ \left( 1-\frac{x}{K} \right) - \frac{\alpha _1\alpha _2}{A_y+x}\frac{x^q}{A_x^q+x^q} - \alpha _2\alpha _3\alpha _4^{-1} \frac{x^q}{A_x^q+x^q} \right\} = 0, \end{aligned}$$

(A.2)

where $\alpha _1=a_y/r_x$, $\alpha _2=a_x/d_y$, $\alpha _3=a_z/r_x$, and $\alpha _4=d_z/r_y$. Eq. (A.2) is used to determine the equilibrium value.

Equating inside the curly bracket with $\alpha _1$ in Eq. (A.2), we obtained

$$\begin{aligned} \alpha _1 = \left( 1-\frac{x}{K} \right) (A_y+x)(A_x^{q}+x^{q})\alpha _2^{-1} x^{-q} - \alpha _3 \alpha _4(A_y+x). \end{aligned}$$

(A.3)

Equation (A.3) was used to produce the bifurcation diagram in Fig. 3.

Fast and slow dynamics

As the values of the parameters $a_x$ and $d_y$ were considerably larger than those of the other parameters, ecotourism was found to have fast-paced dynamics. We obtained an analytical solution for Eq. (A.1b) using the inner approximation method Logan (2013) near $t=0$. Accordingly, Eq. (A.1b), near $t=0$, can be expressed as follows:

$$\begin{aligned} \frac{\mathrm{d}y}{\mathrm{d}t} = \frac{a_x x_0^q}{A_x^q+x_0^q} - d_y y, \end{aligned}$$

(A.4)

where, $x_0=x(0)$. Therefore, the first term on the right-hand side is a constant. If we denote the constant as A, Eq. (A.4) can be easily calculated as follows:

$$\begin{aligned} y_{in}(t) = \frac{A}{d_y}+C_0e^{-d_yt}, \end{aligned}$$

(A.5)

where $C_0$ is an integration constant. Setting $y(0)=0$, we have $y_{in}(t) = A/d_y\left( 1-e^{-d_yt}\right)$, as shown in Fig. 4b in the main text.

If the time is not close to $t=0$, the ecotourism dynamics quickly adapt to the change in slow variables x and z. In this situation, we inserted $y={\bar{y}}$ into Eqs. (A.1a) and (A.1c), where ${\bar{y}}$ is the solution of $dy/dt=0$. Accordingly, the slow dynamics in the model were obtained as follows:

$$\begin{aligned}&\frac{\mathrm{d}x}{\mathrm{d}t} = r_x x \left( 1-\frac{x}{K} \right) - \frac{a_y x}{A_y+x}\frac{a_x d_y^{-1} x^q}{A_x^q+x^q} - a_z xz, \end{aligned}$$

(A.6a)

$$\begin{aligned}&\frac{\mathrm{d}z}{\mathrm{d}t} = r_y \frac{a_xd_y^{-1} x^q}{A_x^q+x^q} - d_z z. \end{aligned}$$

(A.6b)

These dynamics are in agreement with the full dynamics, as shown in Fig. 4c in the main text.