Modelling the Impact of Human Population and Its Associated Pressure on Forest Biomass and Forest-Dependent Wildlife Population

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Introduction
Forests play a significant role in maintaining the ecological balance of the ecosphere [1].Forest maintains microclimate and water cycle [2], prevents soil erosion, and sequesters carbon dioxide from the atmosphere [3], as well as the source of wood, energy, medicine, and fodder [4].The World Bank estimates that about one billion people worldwide depend on the forest as a primary source of medicine [5].Furthermore, forests provide habitat for 80%-90% of the terrestrial biodiversity [6].The highest ratio of endemics per area of all biodiversity hotspots is found in the forests of the eastern arc of mountains in Tanzania and Kenya [7]; it occupies an area of 2000 km 2 and offer habitat to about 121 endemic vertebrate species [5].
Despite the significant roles of the forest, the resources are under threat due to unrelenting pressure from the human population and its associated activities [8].Human population growth has been shown to be significant around the world [9] putting pressure on forest resources [10].The need for food and settlement to accommodate the rapidly growing human population is the major cause of deforestation and the destruction of wildlife habitats [11].The challenge is exacerbated by poverty in developing countries such as Tanzania where the livelihoods of people highly depend on land and forest resources [12].In particular, Shadrack et al. [13] reported that 90% of the households in Tanzania depend on wood charcoal and firewood as their main sources of energy.It is fairly to say if nothing is done to reduce the dependence of forest resource, Tanzania forest land is at risk of disappearing.Similarly, The State of the World's Forests 2020 (FOSO) shows that forest land covers about 31% of the total world's land [14]; nevertheless, between 2010 and 2015, the world experienced a loss of 3.3 million hectares of forest areas [15].Furthermore, the estimate shows that 137 species are lost every single day due to deforestation [16].This means threat to forests is also a threat to biodiversity hotspots.
According to Duncan [17], consumption of forest resources depends on social economic status, human lifestyles, and industrial and agricultural structures, as well as population (human and wildlife) levels.In Africa, social economic status is contemplated as the most significant indirect cause of deforestation [18].A large proportion of African households depend on forest resources for their livelihoods [13], as well as agricultural practices are inefficient due to the lack of advanced technologies [18], thus, putting more pressure on forest lands.Bhardwaj [18] reported that economic growth affects deforestation into two possibilities, either positive or negative.This phenomenon is hypothesized under the Immiserization Theory.The theory postulates that poor people tend to extract more natural resources to meet their needs, leading to forest degradation [19].The theory is supported by the findings from Raphael et al. [20] that assets poor households depend more on forest-generated income than their counterparts.Conversely, economic growth prevents deforestation through the creation of offfarm employment and empowerment of the society to develop better forest management strategies and create environmental awareness for forest conservation [20].Globally, extreme poverty has rapidly declined, World Bank estimates show that the number of people who live on 1.90 USD or less a day worldwide has dropped from 1.9 billion in 1990 to 736 million in 2015 [21].Yet, in Sub-Saharan Africa, the number of people living in extreme poverty is increasing rapidly.The forecast shows that by 2030, 9 in 10 extremely poor people will be living in Sub-Saharan Africa [21], meaning increasing threats to Sub-Saharan Africa forests.In Africa, particularly in Tanzania, the rate of poverty reduction does not match the rate of population growth, leading to an increase in the number of households living under absolute poverty [22].The proportion of the poor by geographic domain in Tanzania shows that 84.4% of poor people live in rural areas while 15.7% live in urban areas [23].According to Bhardwaj [18], unsustainable consumption of natural resources contributes to poverty, and consequently environmental degradation.Therefore, studying the impact of population growth on forest resources has multiplied positive effects on finding better ways to reduce poverty and manage environments.
Despite the challenges, forests remain a vital source of livelihood for people in developing countries [24].Therefore, attempts to completely limit peoples' access and utilisation of forest resources are not possible.However, the management of forest resources to manage harvesting and utilisation is feasible.Different scholars ([5, [25][26][27][28][29][30][31][32] and references therein) have studied the impact of population growth and its associated pressure on forest biomass both theoretically and experimentally.These studies try to understand the interaction between human population and forest biomass.The main questions that were posed are as follows: how does the system that governs human-forest-wildlife species interaction behave over time?What are the key parameters that drive the system?How fast can forest resources recover naturally from destruction?What are the measures needed to conserve the resources?And Is the developed model reliable?This review advances knowledge of the use of theoretical approaches under dynamic system models to understand variables and relationships that form complexity in human-forest interactions and make predictions about the behaviour of the system formed.Knowing that it is not possible to include all articles in the review, we used the procedures proposed by Meline [33] to select which studies to be taken into consideration.Particularly, the articles had to meet the following requirements in order to be included: the paper has been published in the journal listed in the Web of Science and the paper used dynamical systems as well as consists at least two of the keywords.
The rest of the review is structured as follows: Section 2 presents a systematic review methodology used; while in Section 3, various mathematical models that meet the inclusion and exclusion criteria are presented and analysed.The discussion and conclusion of what has been drawn from the Section 3 are presented in Section 4.

Materials and Methods
Given the abundance of publications and research interest in this area, the preestablished inclusion and exclusion guidelines suggested by [33] were established to assist in selecting the papers for the review.An in-depth literature search was carried out in order to identify and compile the publications pertinent to this review.The criteria considered in this review were as follows: the article was published in a journal that is listed on the Web of Science; the dynamical models were employed as the mathematical tool, as well as, at least two of the keywords (mathematical modelling, dynamical system, forest biomass, forest-dependent wildlife, and population pressure) appeared in the title or abstract.To accomplish that, the "OR" and "AND" logical operators were used to search the articles in four different search engines, namely, Google Scholars, Web of Science, Scopus, and ScienceDirect.This comprehensive search turned up a gross list of about 101 papers of which 42 were irrelevant to the focus of this review, leaving 59 articles for the review.Furthermore, the relevant articles were stratified by looking at the state variables used in modelling process, whereby eight groups were formed (Table 1 and Figure 1).The models presented in Subsection 3.1 are representative of each established groups.At least one author of this review thoroughly reviewed each of the selected papers, a number of attributes related to model formulation and model analysis, which are the focus of this review were noted and discussed.

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Journal of Applied Mathematics

Mathematical Models
In the field of forest and wildlife management, mathematical modelling is not a new phenomenon.Management of natural resources such as forests involves many different interrelated variables.These variables and relationships that hold them together form systems that are complex in nature [34].These complexities can be studied and analysed well by using theoretical models.Mathematical models, in particular the use of system of differential equations have been widely employed to investigate systems arising from human-nature interactions and provide predictable solutions [35].Usually, these models use mathematical concepts to explain the system, study the effects of different components, and make predictions about the behaviour of the system [34].
3.1.Dynamical System Models.The dynamical system models are models used to define and predict the interac-tions between several compartments overtime of a phenomenon that is viewed as a system [35].Mathematically, dynamical system can be described as such that Here, X is the set of state variables (compartments) while θ is the set of parameters which govern the links between compartments over time t.
Several researchers employed dynamical system models to study the dynamics of human-forest interaction (e.g., [26,[36][37][38][39][40][41]).This may be explained by the fact that, when human population and its associated pressure increases, the interaction between human population and forest biomass forms a complex system [3].To address this challenge, researchers often use the system of differential equations to explain the system, study the effects of each different variable in the system and make predictions about the behaviour of the system [42].
Bahadur et al. [43], for example, proposed and analysed a dynamical system model (Equation ( 3)) to study the effect of depletion of forest biomass in a forested habitat caused by population pressure (industrialization) on the survival and existence of forest-dependent wildlife species.The model was developed with the assumptions that forest biomass, wildlife species, and population pressure are governed logistically; the density of forest biomass determines the density of wildlife species, and the growth rate coefficient increases as the biomass density of the resources grows; the carrying capacity of wildlife species is affected by the density of forest biomass and the density of industrialization pressure, with the former increasing and the later decreasing.The study suggests that, as the population pressure (industrialization) increases without control, the density of forest biomass decreases, leading to the decreasing of the density of wildlife species and its subsequent extinction.The analytical analysis provides important parameter threshold for the coexistence of all compartment involved in the interactions.This information is useful in policy formulation and implementation.However, the human population, which has a direct impact on forest biomass and wildlife species, was not included in the model formulation, and population pressure was not addressed as cumulative, instead relying solely on industrialization.In line with Bahadur et al. [43], Manju et al. [37] proposed a mathematical model (Equation ( 4)) to study the depletion of forest biomass due to industrialization pressure.The model was formulated based on the assumption that all functions are sufficiently smooth so that all solutions to initial value problems exist uniquely.Further, the densities of forest biomass and industrialization pressure were governed by logistic type equations as well as the forest carrying capacity depends on industrialization.The study considered ratio-dependent interactions with the assumption that in some cases it qualified to represent the reality than linear functional response models.However, the study did not describe precisely which cases ratio-dependent is the best interaction over the linear functional response.
where BðtÞ, WðtÞ, and IðtÞ, respectively, are the density of forest biomass, wildlife species, and industrialization measure.On the other hand, Kumari and Abhinav [27] developed and analysed a model (5) to study the survival of forestdependent wildlife populations in the presence of population pressure-induced mining activities.The study used Hollingtype II functional response to describe the interaction between forest and wildlife species, while the human population was thought to grow logistically.Conversely, population pressure and mining activities were assumed to be governed by mass action functional response.The study is aimed at addressing human-wildlife conflicts, which are the implication of overgrowing mining activities in the forest area.However, the effects of human-wildlife conflicts on human and wildlife populations were not explicitly captured in the model formulation.The model was analysed qualitatively for the boundedness, existence, and stability analysis of the boundary and interior equilibrium points, as well as the uniform persistence of the system.Furthermore, the quantitative analysis was performed to confirm the qualitative analysis, whereby the numerical results demonstrate that the system may not remain persistent if the utilisation rates of the forest are higher than its growth.The study further concluded that, in order to maintain the balance between the populations of humans and wildlife species, forest biomass that is inhabited by wildlife should be used sustainably.
where BðtÞ, W, NðtÞ, PðtÞ, and MðtÞ are the cumulative density of forest biomass, wildlife species, human population, population pressure, and mining activities, respectively.Rachana et al. [40] developed and analysed a mathematical model (Equation ( 6)) to study the effect of deforestation caused by human population and industrialization on forestdependent wildlife species.The growth rate of density of wildlife species that are entirely reliant on forest resources is governed by a logistic model, whose carrying capacity is depleted by human population and industrialization.The stability theory of nonlinear ordinary differential equations was used to study the local and global stability of the interior equilibrium point, as well as the system's persistence.According to findings, deforestation induced by population and industrialization is driving forest resources to extinction and endangering the survival of wildlife species.The numerical simulation was utilized to validate the findings of the analytical analysis, whereby critical parameters of the model were identified.Despite the model incorporates human and wildlife population as state variables, industrialization was treated as the sole source of human population pressure.
4 Journal of Applied Mathematics where BðtÞ, NðtÞ, IðtÞ, and WðtÞ are the cumulative density of forest biomass, human population, industrialization, and wildlife species, respectively.A year later, Kusum et al. [29] developed and analysed a nonlinear mathematical model (Equation ( 7)) to study the effect of deforestation caused by population and population pressure on wildlife species.The model considered a forest habitat where both forest biomass and wildlife species are under threat due to overgrowing human population and their demands.The formulation of the model was based on the assumptions that wildlife species are completely reliant on forestry resources for food and shelter.Because of the desire for agricultural land for food production, wood, space for housing complexes, and economic position, wildlife species are being affected as the human population expands in the forest region.The findings show that the density of wildlife species tends to decline as the parameters responsible for the increase of population pressure increase.Although population pressure affects wildlife populations, the model did not take into account how wildlife population benefits human population.
where BðtÞ, NðtÞ, PðtÞ, and WðtÞ are the cumulative density of forest biomass, human population, population pressure, and wildlife species.
Recently, Rachana et al. [44] developed and analysed a model (Equation ( 8)) to study the impact of human activities on forest biomass and the wildlife population.The study assumed that the growth rate of the wildlife population wholly depends on forest for food and shelter.Further, it was assumed that the growth of forest biomass and human population follows a logistic nonlinear type of equations.The findings show that there is a strong relationship between human activities and forest resources.As to other studies reviewed, Rachana et al.'s [44] study did not include the concept of the residence-based human population which is a critical phenomenon in forest management in developing countries [20].
where BðtÞ, NðtÞ, HðtÞ, and WðtÞ are the cumulative density of forest biomass, human population, human activities, and wildlife species, respectively.
Apart from models that studies depletion of forest biomass and forest-dependent wildlife species, different scholars ( [1,38,45,46]) developed and analysed models for depletion and conservation of forest biomass; the impact of forest depletion on the survival of other biological species was not featured in modelling process.For example, Kumar et al. [38] developed and analysed a model (Equation ( 9)) to study the impact of population and population pressure on forest resources and its conservation by using economic measures.The model was developed with the assumption that the growth of forest biomass and human population were governed by logistic equation, while industrialization, population pressure, and economic measures were governed by mass action functional response.Further, it was assumed that the demands for industrial products increase the intensity 5 Journal of Applied Mathematics of industrialization and eventually destroy forest biomass.Economic measures were therefore assumed to be implemented to lessen the severity of population pressure, which eventually reduce the intensity of industrialization.Although the study incorporates human population as one of the state variables, it does not explore how the decline in forest biomass affects the survival of other biological species.The model analysis suggests that the application of economic measures can significantly reduce population pressure and conserve forest resources if they are applied with enough potential.
where BðtÞ, NðtÞ, PðtÞ, IðtÞ, and EðtÞ are the cumulative density of forest biomass, human population, population pressure, cumulative density of industrialization, and economic measures, respectively.
Other scholars [45,47,48] tried to understand the contribution of environment pollution as the result of human activities on the depletion of forest biomass and the survival of forest-dependent wildlife species.For example, Balram and Jabir [36] developed and analysed a mathematical model (Equation ( 10)) to study the impact of environmental pollution on forest biomass.The model was formulated by assuming that forest biomass is depleted by pollution in a closed region D with a smooth boundary ∂D: The study assumed that the pollutant from the environment does not affect the forest resources directly.Conversely, the uptake pollutant is converted to other toxic substance through some metabolic process; this substance is what affects the growth rate of the forest biomass.The results show that if the time delay in formation of this toxic substance is large enough, then the effects of environmental pollution on forest biomass will be minimized.However, if the rate toxic substance formation is high, then the system will destabilize.
The variables BðtÞ, TðtÞ, UðtÞ, and CðtÞ are the density of forest biomass, concentration of pollutants in the environment, environmental pollution concentration absorbed by the resource biomass at coordinates ðx, yÞ ∈ D and time t ≥ 0 and concentration of toxic substance formed after conversion of pollutant UðtÞ, respectively.

Other Models.
Apart from dynamical models, discrete, stochastic, and data analysis models are also used to study the impact of population and population pressure on biodiversity.For example, Freitas et al. [49] studied the effects of human activities on forest cover dynamics, specifically, they were interested to understand if forest fragmentation, deforestation, and forest regrowth can be defined by topography, roads density, and land use.The generalized least squares regression models were used to fit the data, and the findings showed that road density has a long-term effect on forest cover and are an attractor of topography change.Thus, where the density of roads is high, forests are at risk.They recommended that effective forest management strategies would be well served to consider how roads are distributed.
Vanessa et al. [50] used regression models to investigate the relationship between anthropogenic disturbance and forest characteristics.Fourteen attributes related to population pressure were selected as predictor variables while, seven forest attributes related to species diversity, composition, and structure were response variables.The study backed the evidence that human activities at different spatial scales disrupt the structure and composition of the forest cover.Furthermore, the study shows that the percentage of endangered species is positively correlated to long road networks and logging.
Ali and Maryam [32] developed Multilayer Perceptron Neural Network (MLP) to predict vegetation diversity under ecological conditions and human activities.Mean Absolute Error (MAE) (Equation ( 12)), Mean Square Error (MSE) (Equation ( 13)), and Coefficient of determination (R 2 ) (Equation ( 14)) indices were used to estimate the accuracy and performance of the model [51].
where O i is the observed data, P i is the predicted data, O ave is the mean of the observed data, and n is the number of samples.The MLP model demonstrates the capability of predicting the impact of human activities on vegetation diversity, but with the challenge of identification and quantification 6 Journal of Applied Mathematics of the influential variable.In line with study, Ali et al. [31] developed a model for assessing vegetation density in protected areas which are threatened by tourism activities.
The study used data mining techniques in which the results reveal that the MPL has the ability to predict vegetation density changes under tourism pressure precisely compared to Radial Basis Function Neural Network (RBFNN) and Support Vector Machine (SVM) models.In additional, the model suggested that the area with higher organic matter and soil moisture are likely to tolerate pressure form tourist activities.The use of data mining techniques in this study proved to be the best approach when we are confronting with unavailability of real data.Further, Mauro and Gerardo [52] used quantitative analysis to study the sustainability of the world human population given deforestation stance.
They employed a simplified model based on the stochastic process influenced by a random walk continuous time in combination with a dynamical model that describes human-forest interaction.From the statistical point of view, they showed that, given the growing demands of forest resources, if no action will be taken, the probability that forest resources will accommodate the growing demands of human population is less than 10% in the most optimistic scenario.Yet, the model has limitations due to the use of a simple dynamical model which excludes many variables and parameters, however, it provides a good foundation for incorporating dynamical and stochastic models in addressing environmental issues.Zelin et al. [53] and Isaiah et al. [54] provide an in-depth review of the use of other models in studying human-forest interaction.

Discussion and Conclusion
From this review, it is true that the use of mathematical models specifically the systems of differential equations have continually attracted many researchers in the field of forest management.The growing interest hinges on its potential to untangle the complex phenomenon resulted from humannature interactions and provide tools and information that support the formulation of policies and strategies that aim to conserve forest biomass.Despite the limitations associated with many assumptions in mathematical modelling, especially dynamical system models, through analytical and numerical analysis, the review depicted that, key modelling questions were addressed, that is, the behaviour of the human-forest interaction, what are the parameters that needs attention by policymakers, and the reliability of the developed model.In the same vein, the use of data mining techniques [31,32,52], on the other hand, recently demonstrated its applicability to study the impact of human population and its associated activities on the density of forest biomass in light of the lack of sufficient datasets and unlimited parameters that influence the model outputs.
The natural and social world is dynamic, which is necessitating the need for iterative efforts to find better models to improve forest management.This review identified the potential areas of improving the existed models, the areas ranging from model formulation to model analysis (analytical and numerical).Regarding to model formulation, the review eluci-dates the need for developing a complex and more relevant model to understand and predict the impact of anthropogenic activities on forest biomass and forest-dependent wildlife species under the current growing population pressure, changing climatic conditions and socioeconomic situations.It is believed that the households' income determines the consumption patterns of the forest resource and human lifestyles [20].On the other hand, there is unequal distribution of per capital income between rural areas and urban, where rural areas account for more poor households than urban areas [23].To address the role of socioeconomic aspects on forest management, the developed models must incorporate partition of human populations (rural and urban) variable to capture important information regarding the conservation or forest.Partitioning of human population will accommodate emerging socioeconomic and ecological variables that hold together the human-nature interaction and relationships.
Regarding model analysis, the theory of Lyapunov's function was used to establish global asymptotic behaviour of the system in most of the reviewed works.This approach has a shortfall as there is no systematic way to choose proper candidate of the Lyapunov function [55], its effectiveness is mostly depend on trial and error as well as specific problems.The researchers believe that there is a potential to use the Volterra-Lyapunov stable matrices [56] method especially for models with few number of variables.The method incorporates the Volterra matrix theory into the Lyapunov functions which avoids the necessity to determine the coefficients of the Lyapunov functions under specific conditions [55].Furthermore, mathematical models are known to have input parameters which are not known and with sufficient degree of uncertainty due to natural variation and errors in measurement and parameter estimation.It is equally important to perform uncertainty quantification of the model input parameters to quantify the uncertainty and identify which parameters are more sensitive to the model output.The reviewed works used either single parameter or local sensitivity methods to perform uncertainty quantification.These methods keep all other parameters fixed at default values which makes the methods inadequately in measuring the uncertainty and sensitivity of the input parameter [57].Therefore, there is a potential of using a multidimensional parameter space uncertainty quantification methods such as Latin Hypercube Sampling and Partial Rank Correlation Coefficient (LHS/PRCC) and Extended Fourier Amplitude Sensitivity Test (eFAST) [58].These methods are correlational and variance-based approach, respectively, and have recently proved to have the ability to perform global uncertainty quantification where by all uncertainty can be identified and quantified while all parameters are simultaneously varied.

Table 1 :
Groups of relevant articles based on the state variables used in modelling process.BðtÞ, and PðtÞ are the cumulative density of wildlife species, the forest biomass, and population pressure, respectively.