Role of ecotourism in conserving forest biomass: A mathematical model

Rachana Pathak; Archana Singh Bhadauria; Manisha Chaudhary; Harendra Verma; Pankaj Mathur; Manju Agrawal; Ram Singh

doi:10.1515/cmb-2022-0153

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Role of ecotourism in conserving forest biomass: A mathematical model

Rachana Pathak , Archana Singh Bhadauria , Manisha Chaudhary , Harendra Verma , Pankaj Mathur , Manju Agrawal and Ram Singh

Published/Copyright: July 13, 2023

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From the journal Computational and Mathematical Biophysics Volume 11 Issue 1

Abstract

Ecotourism is a form of tourism involving responsible travel to natural areas, conserving the environment, and improving the well-being of the local people. Its purpose may be to educate the traveler, to provide funds for ecological censervation, to directly benifit the economic development, and political empowerment of local communities. Ecotourism has come up as an important conservation strategy in the tropical areas where diversity of species and habitats are threatened because of the traditional forms of development. This study deals with a non-linear mathematical model with a novel idea for sustainable development of biomass with ecotourism which is imperative in the present scenario. Stability and bifurcation analysis of the model is done and it is observed from our study that the system predicts unstability and exhibits bifurcation if ecotourism goes beyond a threshold value.

Keywords: stability; equilibrium; bifurcation; threshold value

MSC 2010: 34A12; 34A34; 34C15

1 Introduction

Ecotourism plays an important role to increase the economy and conserve biomass, specially in developing countries like India. Tourism is more than movement of people from one place to another for some purposes, it is the agglomeration of services and activities that can impact the biomass in a sustainable way. Tourism in sensitive areas or ecosystems without proper planning and management can harm the biomass and the species because infrastructure is needed to sustain tourism and setting up of the infrastructure requires deforestation and other activities that disturb the ecosystem. On the other hand, thousands of conservation initiatives throughout the world have used ecotourism for benefiting rural communities by providing jobs and money to the inhabitant population of the communities [4].

With the rising human population, it has become important to determine the ways ecotourism can be served as saviour for the biomass and as a fast-evolving sector in tourism industry [6,8,7,9,20]. An ecotourist-driven economy offers a sustainable future for local communities while maintaining ecosystem integrity [1,19]. A significant amount of interest and controversy emerged out of the sustainable development and eco-development literature of the 1970s and 1980s [2,10]. Different types of trends such as photography, bird watching, science research, and tracking incorporates ecotourism which can also be viewed as a subset of natural tourism activities, and can lead to increased incentive for conservation of natural environment [12]. As per International Ecotourism Society (TIES 2015), ecotourism has led to an improvement of the well-being of local communities and helped in educating and imparting satisfactory learning experiences to the tourists [14]. Lockdown imposed by various countries during the COVID-19 pandemic has affected several sectors including ecotourism and pushing millions of people into extreme poverty [3,4,13]. Hence, now it has become mandatory to create a balance between maintaining the diversity of the environment and achieving economic benefits through ecotourism [18].

Hence, with this objective in mind we have framed a mathematical model for ecotourism in protected areas (PAs) by considering three state variables, associated with ecotourism ( T ) , biomass ( B ) , and industry ( I ) . The primary aim of this study is to showcase the significance of ecotourism for a sustainable biomass and to predict the situation when ecotourism increases its threshold value. We consider a mutualism model to study the dynamics of the system. Initially, phenomenological descriptions of interactions were used by the researchers to model the dynamics of mutualisms. Rai et al. [17] considered a mutualist model containing two predators competing for the same prey and also in competition with each other and a mutualist to the prey where the mutualist possesses defensive mechanisms against both the predators. Singh et al. [22] proposed and analysed a system of four autonomous differential equations as a model of four interacting populations, with two species competing with each other for a single prey species in beneficial interaction with a mutualist. Changing the signs of the competition coefficients in the prey-predator (Lotka-Volterra) model is the most well-known example to reflect the positive contribution of mutualism as discussed in previous literature [15,16,23]. We have incorporated the effect of mutualism in ecotourism by considering mutualistic relation between the tourism and biomass which is the novel feature of our model since mutulistic relation between tourism and biomass has not been considered in the previous litreture to the best of our knowledge.

This article is organised as follows: In Section 2, the mathematical model of the problem is fomulated. Section 3 is devoted to the study of boundedness of the system. Basic mathematical results including the existence of equilibrium points, their local stability and the Hopf bifurcation are studied in Section 4. Section 5 includes some results used throughout the research study. Numerical results are obtained in Section 6 to validate our analytical findings from previous sections. Finally, we present the discussions and conclusionin Section 7.

2 Mathematical model

We formulate a nonlinear system of ordinary differential equations considering the interactions among the ecotourism T , biomass B , and industry I . Industries are set up in the PAs to produce accommodation and entertainment facilities to increase the tourism in the area. Keeping all these things in mind the mathematical model is given as follows:

(1) d T d t = T 1 − T L + lB ,

d B dt = rB 1 − B K 1 ( T ) − α BI ( a + B + mT ) ,

d I d t = β BI ( a + B + mT ) − β 0 I ,

where we define K 1 ( T ) = K 0 + KT , T ( 0 ) > 0 , B ( 0 ) > 0 , and I ( 0 ) > 0 . Here we have the following assumptions:

The growth rate of ecotourism is of logistic type.
The ecotourism acts like mutualist in the model.
Interaction between biomass and industrilisation in the absence of ecotourism is Holling type-II functional response.

Where t represent the time, L and K are the carrying capacities of ecotourism and biomass, respectively, r being the intrinsic growth rate of biomass, l and m are mutualism constants, α is the rate of exploitation of biomass for industrialisation, β is the growth rate of the the industrialisation due to biomass, and β 0 is the control rate over industrialisation. The above assumptions are ecologically reasonable and exemplified. a is the half saturation constant.

3 Boundedness

In theoretical biology, boundedness of a system implies that the system is biologically well behaved. The following theorem ensures the boundedness of the system (1).

Theorem 1

The set Ω = ( T , B , I ) : 0 ≤ T ≤ L + l K ∼ , 0 ≤ B ≤ K ∼ , 0 ≤ B + I ≤ ( r + β 0 ) K ∼ β 0 is a region of attraction for all the solutions initiating in the interior of the positive orthant, where K ∼ = max ( K 1 ( T ) ) .

Proof:

From the second equation of the system (1),

d B d t = r B − B 2 K 1 ( T ) − α BI ( a + B + mT ) ,

d B d t ≤ rB 1 − B K 1 ( T ) .

This implies that

lim t → ∞ sup B ( t ) ≤ K 1 ( T ) , where K ∼ = max ( K 1 ( T ) ) .

Thus,

B max = K ∼ .

The first equation of the system (1) gives,

d T d t = T − T 2 L + lB ,

d T d t ≤ T − T 2 L + l K ∼ .

This implies that

T max = L + l K . ∼

Adding last two equations of the system (1), W = B + I , we have

d W d t = d B d t + d I d t = r B − r B 2 K 1 ( T ) − α BI ( a + B + mT ) + β BI ( a + B + mT ) − β 0 I ,

d W d t ≤ rB − β 0 I .

In real world, growth rate of industrialization due to biomass is always less than exploitation rate of biomass. So, β < α .

Thus,

dW dt ≤ ( r + β 0 ) K ∼ − β 0 W ,

W max = ( r + β 0 ) K ∼ β 0 .

Hence, all the solutions of (1) enter into the region: Ω = ( T , B , I ) : 0 ≤ T ≤ L + l K ∼ , 0 ≤ B ≤ K ∼ , 0 ≤ B + I ≤ ( r + β 0 ) K ∼ β 0 .

Hence proved.

4 Basic mathematical results

4.1 Equilibrium analysis

The system (1) has six nonnegative equilibrium points. The equilibrium points E 0 ( 0 , 0 , 0 ) , E 1 ( 0 , K 0 , 0 ) , E 2 ( L , 0 , 0 ) , E 3 L + l K 0 + KL 1 − Kl , K 0 + KL 1 − Kl , 0 with conditions Kl < 1 , E 4 ( 0 , B ̄ , I ̄ ) , where B ̄ = β 0 a ( β − β 0 ) , I ̄ = r 2 1 − B ̄ K 0 [ a + B ̄ ] with conditions β > β 0 and β 0 a ( β − β 0 ) K 0 < 1 , always exist. Now we can show the existence of interior equilibrium point E * ( T * , B * , I * ) as follows:

The existence of E * ( T * , B * , I * ) .

Here T * , B * , and I * are the positive solutions of the system of algebraic equations given below

T * 1 − T * L + l B * = 0 ,

r * B 1 − B * K ( T * ) − α B * I * ( a + B * + m T * ) = 0 ,

(2) β B * I * ( a + B * + m T * ) − β 0 I * = 0 .

After solving the system, we get T * = L + l B * , B * = β 0 ( a + mL ) β − β 0 ( 1 − ml ) , I * = r 2 1 − B * ( K 0 + KL + Kl B * ) [ a + mL + ( 1 + ml ) B * ] are always positive solutions of the system of equation (2) with conditions β > β 0 ( 1 + ml ) and B * ( K 0 + KL + Kl B * ) < 1 .

4.2 Local stability

The local stability of system (1) is discussed as follows:

J ( E ) = e 11 e 12 0 e 21 e 22 e 23 e 31 e 32 e 33 .

Coefficients of the Jacobian matrix of the system (1) at the equilibrium points are as follows:

e 11 = 1 − 2 T L + lB , e 12 = l ( T ) 2 ( L + lB ) 2 , e 21 = r ( B ) 2 K ( K 0 + KT ) 2 + α BIm ( a + B + mT ) 2 ,

e 22 = r − 2 rB ( K 0 + KT ) − a α I + m α T I ( a + B + mT ) 2 , e 23 = − α B ( a + B + m T ) , e 31 = β B I m ( a + B + m T ) 2 ,

e 32 = a β I + m β TI ( a + B + mT ) 2 , e 33 = β B ( a + B + mT ) − β 0 .

Accordingly, the linear stability analysis about the equilibrium points E i , i = 0,1,2,3,4 and E * gives the following results:

The equilibrium point E 0 is unstable manifold in T − B plane and E 1 is unstable in T direction.

The equilibrium point E 2 is unstable manifold in B direction , E 3 is unstable manifold in I direction and E 4 is unstable in T direction. The stability behaviour of equilibrium point E * ( T * , B * , I * ) is not obvious. The characteristic polynomial of the Jacobian matrix at the equilibrium point E * ( T * , B * , I * ) is given by λ 3 + C 1 λ 2 + C 2 λ + C 3 = 0 , where C 1 = − ( a 11 + a 22 + a 33 ) , C 2 = a 11 a 22 + a 11 a 33 + a 22 a 33 − a 12 a 21 − a 23 a 32 , C 3 = a 11 a 23 a 32 + a 12 a 21 a 33 − a 12 a 23 a 31 − a 11 a 22 a 33 . By Routh-Hurwitz criteria, equilibrium point E * ( T * , B * , I * ) is locally asymptotically stable if C 1 , C 3 > 0 and C 1 C 2 > C 3 , otherwise it is unstable.

4.3 Hopf bifurcation

Here we establish conditions that guarantee the occurrence of a Hopf bifurcation [5,11,21,24] near the positive equilibrium point E * .

Theorem 2

When carrying capicity K of the biomass B crosses the critical value K cr , the model system undergoes Hopf bifurcation around the positive equilibrium E * under the following conditions:

(3) C 1 ( K cr ) > 0 , C 3 ( K cr ) > 0 , C 1 ( K cr ) C 2 ( K cr ) − C 3 ( K cr ) = 0 ,

and

(4) [ C 1 ( K ) C 2 ( K ) ] K = K cr ′ − C 3 ′ ( K cr ) ≠ 0 .

Proof

We choose K as the bifurcation parameter and investigate if there exists a critical value K = K cr such that conditions (3) and (4) are satisfied. For the occurrence of Hopf bifurcation at K = K cr , the characteristic equation of the Jacobian matrix J ( E * ) must be of the form□

(5) ( λ 2 ( K cr ) + C 2 ( K cr ) ) ( λ ( K cr ) + C 1 ( K cr ) ) = 0 ,

which has roots λ 1 ( K cr ) = i C 2 ( K cr ) , λ 2 ( K cr ) = − i C 2 ( K cr ) , λ 3 ( K cr ) = − C 1 ( K cr ) < 0 , then obviously, C 3 ( K cr ) = C 1 ( K cr ) C 2 ( K cr ) . To show that the Hopf bifurcation occurs at K = K cr , we need to validate the transversality condition:

(6) d ( Re λ j ( K ) ) d K K = K cr ≠ 0 , j = 1 , 2 .

On substituting λ j ( K ) = u ( K ) + iv ( K ) in equation (5) and calculating the derivative, we have

(7) E ( K ) u ′ ( K ) − F ( K ) v ′ ( K ) + M ( K ) = 0 ,

(8) F ( K ) u ′ ( K ) + E ( K ) v ′ ( K ) + N ( K ) = 0 ,

where

E ( K ) = 3 u 2 ( K ) − 3 v 2 ( K ) + C 2 ( K ) + 2 C 1 ( K ) u ( K ) ,

F ( K ) = 6 u ( K ) v ( K ) + 2 C 1 ( K ) v ( K ) ,

N ( K ) = 2 u ( K ) v ( K ) C 1 ′ ( K ) + C 2 ′ ( K ) v ( K ) ,

and

M ( K ) = C 2 ′ ( K ) u ( K ) + u 2 ( K ) C 1 ′ ( K ) − v 2 ( K ) C 1 ′ ( K ) + C 3 ′ ( K ) .

Here at K = K cr , u ( K cr ) = 0 and ( K cr ) = C 2 ( K cr ) .

Hence, we have

E ( K cr ) = − 2 C 2 ( K cr ) ,

F ( K cr ) = − 2 C 1 ( K cr ) C 2 ( K cr ) ,

N ( K cr ) = C 2 ′ ( K cr ) C 2 ( K cr ) ,

and

M ( K cr ) = C 3 ′ ( K cr ) − C 2 ( K cr ) C 1 ′ ( K cr ) .

On solving for u ′ ( K cr ) from equations (7) and (8), we have

d Re ( λ j ( K ) ) d K K = K cr = u ′ ( K cr ) ,

= − M ( K cr ) E ( K cr ) + N ( K cr ) F ( K cr ) E 2 ( K cr ) + F 2 ( r cr ) ,

= C 3 ′ ( K cr ) − C 2 ( K cr ) C 1 ′ ( K cr ) − C 1 ( K cr ) C 2 ′ ( K cr ) 2 ( C 2 ( K cr ) + C 1 2 ( K cr ) ) ≠ 0 ,

provided d d K [ C 1 ( K ) C 2 ( K ) − C 3 ( K ) ] K = K cr ≠ 0 . Hence, the transversality condition holds, which implies that a Hopf bifurcation occurs at K = K cr .

5 Numerical simulations

Now we explore the dynamics of our model (1) with the help of numerical tools and techniques. For numerical simulations, we use the following parameter values as default:

(9) L = 200 , l = 0 . 002 , α = 4 , β = 3 , β 0 = 0 . 8 , m = 8 , a = 4 , K = 8 , K 0 = 800 , and r = 0 . 2 ,

and change it accordingly to determine the effect of various parameters on the system dynamics. We also keep the initial condition fixed at ( 10 , 10 , 10 ) unless otherwise stated.

For the above set of parameters, the condition of existence of equilibrium E * and stability condition at E * in table are satisfied. The eigenvalues of the Jacobian matrix corresponding to the equilibrium point E * for the model system (1) are − 1.00125 , − 0.00355159 + 0.298587 i , and − 0.00355159 − 0.298587 i for the given set of parameters. It is observed that all the eigenvalues of Jacobian matrix corresponding to the point E * are negative. Hence, the equilibrium point E * is asymptotically stable for these set of values. Further, it is found that all other equilibrium points are unstable for the above set of parameters.

Figure 1 shows that the system is aymptotically stable for the above set of parameters. Now, keeping all the parameters fixed and varying K we observe that the biomass exhibits stable behaviour till K = 8 as shown in Figure 2(a). Further, stable phase portrait is obtained between biomass and industry in Figure 2(b). On increasing the value of K from K = 8 , the system starts exhibiting unstabilty and prominant oscillating nature of biomass with time is observed at K = 12 . Time series plot of biomass with time at K = 12 is displayed in Figure 3(a). Phase portrait between biomass and industry is drawn in Figure 3(b). Thus, numerically, it is observed that K acts as a bifurcation parameter in the system. The system shows stable behaviour till K ≤ 8 and as the value of K is increased from 8 , the system starts oscillating and at K = 9.5 , it shows hopf bifurcation. Hopf bifurcation in the system for K = 9.5 can be easily seen in Figure 4.

Figure 1

Variation of all the system variables with time.

Figure 2

Stable nature of the system for K = 8 . (a) Variation in biomass with time and (b) phase portrait of the system for K = 8 .

Figure 3

Unstable nature of the system for K = 12 . (a) Variation in biomass with time and (b) phase portrait of the system for K = 12 .

Figure 4

Bifurcation plot with K as a bifurcation parameter.

Moreover, we observe that the system exhibits bifurcation behaviour for two other parameters as well, β and β 0 . In Figure 5(a), we observe that the system shows oscillatory behaviour when β 0 = 0.5 . Further, Figure 5(b) displays the plot between biomass and β 0 representing the bifurcation point at the value β 0 = 0.5 . In addition, we draw Figure 6 showing the bifurcation plot with respect to parameter β 0 at 0.5 .

$Figure 5 β 0 {\beta }_{0} as a bifurcation parameter. (a) Oscillatory nature of biomass for β 0 = 0.5 {\beta }_{0}=0.5 and (b) variation in biomass with β 0 {\beta }_{0} .$

Figure 5

β 0 as a bifurcation parameter. (a) Oscillatory nature of biomass for β 0 = 0.5 and (b) variation in biomass with β 0 .

$Figure 6 Bifurcation plot with β 0 {\beta }_{0} as a bifurcation parameter.$

Figure 6

Bifurcation plot with β 0 as a bifurcation parameter.

Similary, Figure 7(a) is a time series graph exhibiting the oscillatory nature of the system at β = 4.2 with other parameters fixed and Figure 7(b) shows the plot between biomass and β exhibiting the bifurcation in the system at β = 4.2 . Further, Figure 8 shows the bifurcation plot with respect to parameter β at 0.6 .

$Figure 7 β \beta as a bifurcation parameter. (a) Oscillatory nature of biomass for β = 4.2 \beta =4.2 and (b) variation in biomass with β \beta .$

Figure 7

β as a bifurcation parameter. (a) Oscillatory nature of biomass for β = 4.2 and (b) variation in biomass with β .

$Figure 8 Bifurcation plot with β \beta as bifurcation parameter.$

Figure 8

Bifurcation plot with β as bifurcation parameter.

Thus, numerical simulation of the system confirms the existence of three parameters K , β , and β 0 at which hopf bifurcation occurs.

6 Conclusion

The present study proposes the benefits of conserving biomass from ecotourism. To model the system, three system variables, ecotourism, biomass, and industry are considered to form a system of three dimensional nonlinear ordinary differential equations. Boundedness of the system and equilibrium analysis is done. Conditions for the local stability of interior equilibrium point are obtained. These conditions confirm that the system with all the three variables, if exists, will attain stability if certion conditions are satisfied. Moreover, we have proved that as carrying capicity K of the biomass B crosses the critical value K cr , the model system undergoes Hopf bifurcation around the positive equilibrium E * under some conditions. Numerical simulation of the system is performed for a set of parameters. It is observed that all the conditions obtained from analytical study are satisfied numerically. We have obtained the following important findings in our paper:

Numerical value of bifurcation point of the bifurcation parameter K is determined and it is showed that system exhibits Hopf bifurcation at K = 9.5 . It implies that the system gets disturbed and starts showing unstable behaviour if carrying capacity of biomass increases above a certion level as it will be difficult to manage tourists arriving due to inceared carrying capacity of biomass.
In addition, rate of exploitation of biomass for industrialisation, β and rate of industrial decline, β 0 are also found to be bifurcation parameters. We observed through numerical simulation that the system exhibits hopf bifurcation for β = 4.2 and β 0 = 0.5 . This implies that system becomes unstable if rate of expoitation of biomass for setting up industries and making industrial products to promote ecotourism goes beyond a certion level. Also, our study reveals that the rate of decline in industialisation should also be optimal otherwise system would become unstable.

Hence, our study establishes the fact that carrying capacity of biomass, rate of exploitation of biomass for industrialisation, and rate of industrial decline must be kept at an optimal level to ensure stability in the system, otherwise system will become unstable.

Funding information: This research received specific grant from Research and Development Project, UP Government via letter number 47/2020/606/70-42021-04(56)/2020.
Conflict of interest: The authors have no conflicts of interest to disclose.
Ethics Statement: This research does not require ethical approval.

References

[1] Aliani, H., Kafaky, S. B., Monavari, S. M., & Dourani K. (2018). Modeling and prediction of future ecotourism conditions applying system dynamics. Environmental Monitering and Assessment, 190, 729. doi: 10.1007/s10661-018-7078-4.Search in Google Scholar PubMed

[2] Blersch, D., & Kangas, P. (Feb. 2013). A modeling analysis of the sustainability of ecotourism in Belize. Environment, Development and Sustainability: A Multidisciplinary Approach to the Theory and Practice of Sustainable Development, 15(1), 67–80. Springer.10.1007/s10668-012-9374-4Search in Google Scholar

[3] Buckley, R. (2021). Pandemic travel restrictions provide a test of net ecological effects of ecotourism and new research opportunities. Journal of Travel Research, 60(7), 1612–1614.10.1177/0047287520947812Search in Google Scholar

[4] Cherkaoui, S., Boukherouk, M., Lakhal, T., Aghzar, A., & El Youssfi, L. (2020). Conservation amid COVID-19 pandemic: Ecotourism collapse threatens communities and wildlife in Morocco. E3S Web of Conferences, 183, 01003. EDP Sciences.10.1051/e3sconf/202018301003Search in Google Scholar

[5] Deng, L., Wang, X., & Peng, M. (2014). Hopf bifurcation analysis for a ratio-dependent predator-prey system with two delays and stage structure for the predator. Applied Mathematics and Computation, 2014, 214–230.10.1016/j.amc.2014.01.025Search in Google Scholar

[6] Dubey, B., & Hussain, J. (2004). Models for the effect of environmental pollution on forestry resources with time delay. Nonlinear Analysis: Real World Applications, 5(3), 549–570. doi: 10.1016/j.nonrwa.2004.01.001.Search in Google Scholar

[7] Dubey, B., & Narayanan, A. S. (2010). Modelling effects of industrialization, population and pollution on a renewable resource. Nonlinear Analysis: Real World Applications, 11(4), 2833–2848. doi: 10.1016/j.nonrwa.2009.10.007.Search in Google Scholar

[8] Dubey, B., Sharma, S., Sinha, S., & Shukla, J. B. (2009). Modelling the depletion of forestry resources by population and population pressure augmented industrialization. Applied Mathematical Modelling, 33(7), 3002–3014. doi: 10.1016/j.apm.2008.10.028.Search in Google Scholar

[9] Dubey, B., Upadhyay, R. K., & Hussain, J. (2003). Effects of industrialization and pollution on resource biomass: A mathematical model. Ecological Modelling, 167(1–2), 83–95. doi: 10.1016/S0304-3800(03)00168-6.Search in Google Scholar

[10] Fennell, D. A. (2009). Ecotourism. In N. Thrift & R. Kitchin (Eds.), The encyclopaedia of human geography (Vol. 11, pp. 197–205). Oxford: Elsevier.Search in Google Scholar

[11] Hassard, B. D., Kazarinoff, N. D., & Wan, Y. H. (1981). Theory and application of Hopf Bifurcation. Cambridge: Cambridge University Press.Search in Google Scholar

[12] Kim, S. R., & Lee, S. D. (May, 2018). An analysis of tourist participation restoration-ecotourism through systems thinking. IOP Conference Series: Earth and Environmental Science, 151(1), 012015. IOP Publishing.10.1088/1755-1315/151/1/012015Search in Google Scholar

[13] Laudari, H. K., Pariyar, S., & Maraseni, T. (2021). COVID-19 lockdown and the forestry sector: Insight from Gandaki province of Nepal. Forest Policy and Economics, 131, 102556.10.1016/j.forpol.2021.102556Search in Google Scholar PubMed PubMed Central

[14] Lee, T. H., & Jan, F. H. (2018). Ecotourism behavior of nature-based tourists: An integrative framework. Journal of Travel Research, 57(6), 792–810.10.1177/0047287517717350Search in Google Scholar

[15] May, R. M. (1981). Theoretical ecology: Principles and applications (2nd ed.). Oxford: Oxford University Press.Search in Google Scholar

[16] Misra, A. K., & Lata, K. (2013). Modeling the effect of time delay on the conservation of forestry biomass. Chaos, Solitons & Fractals, 46, 1–11. doi: 10.1016/j.chaos.2012.10.002.Search in Google Scholar

[17] Rai, B., Khare, P., & Singh, M. (2013) Mutualistic interactions leading to coexistence in competitive ecological communities: Mathematical approach and simulation studies. Journal of International Academy of Physical Sciences, 17(4), 325–354.Search in Google Scholar

[18] Rilov, G., Fraschetti, S., Gissi, E., Pipitone, C., Badalamenti, F., Tamburello, L., … Benedetti‐Cecchi, L. (2020). A fast-moving target: Achieving marine conservation goals under shifting climate and policies. Ecological Applications, 30(1), 02009.10.1002/eap.2009Search in Google Scholar PubMed PubMed Central

[19] Scheyvens, R. (1999). Ecotourism and the empowerment of local communities. Tourism Management, 20(2), 245–249. doi: 10.1016/S0261-5177(98)00069-7.Search in Google Scholar

[20] Shukla, J. B., Misra, O. P., Agarwal, M., & Shukla, A. (1988). Effect of pollution and industrial development on degration of biomass-resource: A mathematical model with reference to Doon Valley. Mathematical and Computer Modelling, 11, 910–913. doi: 10.1016/0895-7177(88)90626-7.Search in Google Scholar

[21] Song, Y. L., & Wei, J. J. (2004). Bifurcation analysis for Chen’s system with delayed feedback and its application to control of chaos. Chaos Solitons Fractals, 22, 75–91.10.1016/j.chaos.2003.12.075Search in Google Scholar

[22] Tripathi, S., Mishra, R. C., Singh, B., & Singh, M. (2022). Ordinary differential equation models in population ecology, with special reference to deterrence. GANITA, 72(1), 249–261 249.Search in Google Scholar

[23] Vandermeer, J., & Boucher, D. H. (1978). Varieties of mutualistic interaction in population models. Journal of Theoretical Biology, 74, 549–558.10.1016/0022-5193(78)90241-2Search in Google Scholar PubMed

[24] Verma, H., Antwi-Fordjour, K., Hossain, M., Pal, N., Parshad, R. D., & Mathur, P. (2021). A “Double” fear effect in a tri-trophic food chain model. The European Physical Journal Plus, 136, 905. doi: 10.1140/epjp/s13360-021-01900-3.Search in Google Scholar

Received: 2023-02-20

Revised: 2023-04-26

Accepted: 2023-05-13

Published Online: 2023-07-13

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https://doi.org/10.1515/cmb-2022-0153

Keywords for this article

stability; equilibrium; bifurcation; threshold value

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