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In a region under consideration, let at any instant (t >0), x and y represent the prey and predator population densities, respectively. The rate of change of each model species density at time t is made on the following assumptions:

1.

Prey population grow logistically in the absence of predator with birth rate r, which is affected by the fear ((f_1)) of predator (when predators are around).

2.

There is a reduction in the rate of prey density change due to three types of death, namely, natural death with the rate (d_1), fear related death5 with the level of fear (f_2) and over crowding death with the rate (d_2).

3.

Also, the rate of change of prey density decreases due to predation of predator population following a predator-dependent functional response describing both predatory and prey schooling behaviors10. Response function is expressed in functional form describing as (zeta (x, y)=frac{cxy}{1+chxy}), where c denotes the rate of consumption and h represents handling time of predator for one prey.

4.

Predator population survive in the system by consuming prey population only. They grow with conversion efficiency (c_1) of prey biomass into predator biomass.

5.

Predator population harvested from the system which reduces its rate of density. We consider a nonlinear harvesting term (Michaelis-Menten type) given by, (H(y)=dfrac{qEy}{p_1E+p_2y}). Here, parameters q and E, respectively, represent the catchability rate and harvesting effort. It is easy to observe that (Hrightarrow frac{q}{p_1}y) as (Erightarrow infty) for a fixed value of y. Also, (Hrightarrow frac{q}{p_2}E) as (yrightarrow infty) for a fixed value of E. Therefore, at higher effort levels, (p_1) is proportional to the stock level-catch rate ratio and at higher levels of stock, (p_2) is proportional to the effort level-catch rate ratio.

6.

Lastly, we assume that the predator population experience natural as well as over crowding related death with the rates (d_3) and (d_4), respectively.

Keeping all these above assumptions in mind, we formulate the following prey–predator model:$$begin{aligned} frac{dx}{dt}= & {} frac{rx}{1+f_1y}-(1+f_2y)d_1x-d_2x^2-frac{cxy^2}{1+chxy}nonumber ,\ frac{dy}{dt}= & {} frac{c_1cxy^2}{1+chxy}-d_3y-d_4y^2-frac{qEy}{p_1E+p_2y}. end{aligned}$$
(1)
System (1) is to be analyzed with the initial conditions (x(0),y(0) >0). All the model parameters are assumed to be positive constants and their hypothetical values that we used for numerical calculations are as follows:$$begin{aligned}{} & {} r=3.1, f_1=1, f_2=0.4, d_1=0.1, d_2=0.08, c=0.11, h=0.1, c_1=0.5, d_3=0.1,nonumber \{} & {} d_4=0.06, q=0.65, E=0.5, p_1=0.5, p_2=0.65. end{aligned}$$
(2)
In Table 1, we have provided system’s equilibria, sufficient conditions of their existence and stability. Mathematically, it is difficult to determine the existence of coexistence (interior) equilibrium point(s) by the given nullclines. So, we visualize it numerically (see Fig. 1). It is apparent from the figure that on increasing the value of E, number of coexistence equilibrium points reduces and after a certain range there is no coexistence equilibrium point.Table 1 Sufficient conditions for the existence and stability of different equilibrium points of system (1).Full size tableFigure 1Nullclines for different values of E. Other parameters are same as in (2).Full size image
Transcritical bifurcationFrom Table 1, it is clear that the equilibrium (E_0) is stable if (rr^{TB}=d_1=0.1)) then (E_0) becomes unstable and the equilibrium point (E_1) exists and becomes stable.Figure 2Transcritical bifurcation with respect to r. Rest of the parameters are same as in (2).Full size imageHopf bifurcationOne of the most common dynamics in interacting population dynamics is oscillating behavior, which implies that there is a Hopf bifurcation. By local changes in equilibrium properties, Hopf bifurcation describes when a periodic solution appears or disappears. In this section, we study the Hopf bifurcation through the coexistence equilibrium (E^*) with respect to the model parameter E. Discussion for the existence of Hopf bifurcation is as follows:As it is easy to follow, we verify Hopf bifurcation numerically. We have considered the parameters value same as in (2) except (c=0.1) and E. At (E=E^{[HB]}=0.1196559641), the trace of the Jacobian matrix at (E^*(2.618402886, 2.352228027)) is zero and determinant, (Det(J_{E^*})=0.4474794791 >0). The value of (dfrac{d(Tr(J_{E^*}))}{dE}Big |_{E=E^{[HB]}}=-0.02965188514ne 0). Therefore, the transversality condition for Hopf bifurcation is also satisfied at (E=E^{[HB]}). Thus, these results confirm that the system (1) experiences a Hopf bifurcation2 around (E^*(2.618402886, 2.352228027)).Moreover, we obtain Lyapunov number (L_1=-0.04728284756pi More

The contributions of SMFEs to the local economy and developmentSMFEs are characterized by limited resources, hence their inability to employ more people however, the few being employed to aid in the operations of the businesses contribute to the reduction of the employment gap among the youth in the study areas. The employment opportunities provided by SMFEs supplement the central government’s efforts to offer employment to the people. Subsequently, the people in the area depend on it for their livelihood to improve their living standards. The study found a diverse number of SMFEs in terms of wood and non-wood-related activities (Fig. 3) that people engage in as primary or secondary jobs. Some evidence has proved that SMFE’s contribution to forest employment is above 50% in some countries like Brazil, Uganda, Guyana, and China, and almost 80 to 90% of all forest-based enterprises in most countries7. This may directly impact efforts to reduce poverty by improving the living standards of people who form and operate SMFEs as a livelihood.Zada et al.10 also reported that households who own SMFEs had a wealth index increase from 5.4 to 7.4 whereas those without SMFEs had an index of 4.9. SMFEs do have the potential to improve household income levels which can lead to reinvesting and expansion. This study found that the monthly expenses of SMFEs contribute to 46.2% of their monthly sales. Therefore, if SMFEs can increase significantly, their ability to reinvest while observing the best practices in operating their businesses then they will be able to maximize their turnovers. This can result in expansion and more employment opportunities for others hence, reducing the burden on the government to provide employment.There is a direct positive and significant relationship between SMFEs and local economic development11. SMFEs were reported to positively and significantly mediate the relationships between government support, entrepreneurship knowledge, and local economic development. SMFEs with informal or formal training can ensure government support is efficiently used in tapping into entrepreneurial knowledge to drive their impacts on local economies. This will also allow them to grow into sustainable businesses while also promoting the sustainability of forest resources which they depend on for raw materials.Operational characteristics and impacts of SMFEs on sustainabilityAround the globe and per the laws of Ghana, businesses are required to fulfill certain obligations to enable them to run smoothly12. Failure to undertake these tasks may attract severe penalties, including criminal charges that may carry significant jail terms. An example is failure to pay taxes and adhere to certain regulations. This section looks at certain characteristics of SMFEs in this study that project their impact on sustainable forest management.Firstly, the laws of Ghana make it mandatory for all business categories to pay tax and as such, SMFEs are not left out. However, the major challenge with taxes in Ghana and by extension, the world, is compliance. Mantey13 reported that 59.1% of small business owners did not understand the Ghanaian Tax System. The key lesson drawn from this observation is that, as SMFEs, one of their characteristics is that they generate a smaller income compared to larger companies or corporate bodies. This goes a long way in determining the amount they pay as taxes. In addition, by their nature, they can under-declare the revenue they make to influence the amount they will be taxed. This calls for the development and flawless implementation of mechanisms to monitor and audit these SMFEs to ensure that they comply with tax directives and regulations.Mantey13 further reported that 57.4% of the surveyed business owners are not aware of most tax laws and guidelines on the taxation of incomes for organizations. Some blamed their inability to pay taxes on the business being slow and others were unwilling to give a response to why they were unable to pay their taxes. In this study, majority (77.5%) indicated that they pay taxes. It was also established in this study that payment of taxes has a significantly weak correlation with the educational background of the respondents. Though the majority of the SMFEs paid taxes, it may not be directly linked to all respondents having some form of formal education and vice versa. However, this may be factored in when considering the training and mentoring of SMFEs to contribute to local development by paying their taxes. More SMFEs may endeavor to pay their taxes regularly if they understand what these taxes can do to improve their work environment.Governments in recent times have stepped up revenue mobilization efforts to capture more businesses into the tax bracket of the country. This has seen the revenue authorities recruit and train more revenue officers to reach businesses like SMFEs which are mostly not reachable due to their inability to register their businesses.Secondly, the majority (71.25%) of SMFEs in this study was not registered. This adds to the general belief that most businesses operate without the required licenses or have failed to renew their expired licenses. Some studies also made similar observations and arrived at the lack of enforcement of laws, as a key reason why many businesses in developing countries remained unregistered contrary to the requirements of the law14,15. Further analysis showed SMFEs who belong to associations are likely to register their businesses because it is a requirement to join them. The benefits of belonging to an association include access to loan facilities and other credit programs and therefore some SMFEs do not want to risk missing out through failure to get their business registered16.SMFEs need to get registered for them to be considered legitimate business entities however, this seems to be a challenge in most developing countries. Tomaselli et al.17 found this assertion relevant when investigating SMFEs access to microfinance. Registration of business is a key requirement to access loan facilities and so is belonging to a recognized association. Associations are known to serve as guarantors for members who want loan facilities from banks and other financial institutions to expand their businesses16. Unregistered, unregulated, and unmonitored SMFEs are those whose activities tend to compromise the sustainability of forest resources18. Therefore, registration of SMFEs does that only serve the interest of governments but also the interests of these SMFEs themselves.The third has to do with the sourcing of raw materials. Ghana being a tropical country is blessed abundantly with forest resources but over the decades, the overexploitation of these forests has brought to the brink of extinction, various species of both plant and animal life19. The dependence of SMFEs on the forests cannot be underestimated as literature, citing Osei Tutu et al.20, posits that SMFEs contribute to 95% of the income of some rural households. This study shows that 68.8% of SMFEs get their raw materials directly from the forest. Both woody and non-woody materials are in abundance and can be extracted with minimal cost.In sourcing raw materials from the forests in Ghana, SMFEs are required to obtain permits or licenses from the relevant authorities such as the forestry commission. This permit/license is what allows or gives this SMFEs access to otherwise inaccessible forest reserves to harvest raw materials20. Additionally, these documents can go as far as determining the type and quantity of materials to harvest. It can also determine the type of access granted as these accesses can vary or differ depending on the time or season of harvest18. The issuance of permits and licenses is meant to monitor and regulate resource harvesting with the primary goal of checking the overexploitation of these resources. However, this is not possible due to the high levels of non-compliance by SMFEs21. Evident in this study is the 78.2% of SMFEs who gather raw materials from the forest without permits/licenses.Osei Tutu et al.18 concluded that the neglect of the SMFEs sub-sector is responsible for the loss of state revenue because of their unwillingness to register and pay appropriate taxes and permit fees for their illegal and unsustainable business operations. The report further posits that “despite the numerous support channels (national and international) available to them, the roles played by SMFEs in poverty reduction are significantly unimpactful hence the need to intensify capitalizing on all opportunities to address challenges they present.” The government institutions in charge of these forest resources depend on these permits and license fees to supplement their already insufficient government subventions for the operations. Therefore, losing revenues may undermine their sustainability programs.Driving factors of SMFEsThe ability of a business to thrive highly depends on its ability to overcome certain challenges within its operating environment22. That alone, however, is not enough as certain factors ignite the ambition of a business. These factors decisively influence the success or the failure of the business hence, they are identified as determinants. The study sought to identify some determinants that drive the activities of SMFEs. Responses from the SMFEs concluded that economic and social factors such as resource availability, profits/revenue, employment, and labor are the key determinants that drive the SMFEs.Resource availability was the major driver of their activities cited by 91.3% of SMFEs. This is because, the numerous forests the nation is endowed with provide abundantly, the raw materials needed for them to use. Due to the favorable climatic conditions prevailing in the high forest zones, there is a constant supply of materials needed by SMFEs to produce their products for business23. In addition, availability means less competition for limited resources and therefore it boils down to the ability to process these raw materials into finished goods for market consumption hence, reducing the costs of production24.SMFEs also pointed to profits/revenue, as the factor driving their activities to engage in, and sustain their business. The abundance and readily availability of raw materials are very important to the growth of their business and in turn, help them maximize their returns. This is because the inputs they make to acquire the raw materials are relatively low in comparison to the total revenues they generate. This observation is also reflected in the captured expenditures they make as inputs or investments into their businesses.SMFEs that need technologically advanced mechanisms and equipment are those that are required or inclined to make heavy investments whereas those that need simple tools and equipment invest less. Whichever the case, the nature of SMFEs suggests that a business that requires raw materials with very minimal or no costs involved at all, yet yields very high profits, is how people can improve their living25. Badini et al.26 classified enabling environment of SMFEs into external and internal factors where financial capital, business management, and organizational capacities form internal factors. On the other hand, external factors include regulatory frameworks, forest law enforcement, and natural capital which refers to the stock of natural resources or environmental assets. The success of any SMFE is largely dependent on these factors.Finally, 8.75% of the SMFEs view labor and employment, as the determinants driving their existence. For them, compared to other labor-intensive ventures, their business does not require huge labor to get work done. The few hands needed means most of the revenues do not go to paying workers. They can dictate and bargain to their advantage because there are many people without jobs hence a job turned down, because of less encouraging benefits is gladly accepted by another25. Ultimately, the study finds that labor is cheap in some areas of the SMFEs’ environs primarily, due to unemployment.Sustainability challenges in forest management relative to SMFEs activitiesSince the United Nations Conference on Environment and Development (UNCED) in Rio de Janeiro, Brazil in 1992, key challenges of SFM have broadly covered the sustainability of forest resources through the reduction of deforestation and forest degradation, conservation and protection of biological diversity, genetic resources sustainability and improving forest goods and services valuation27. It is important to note that SMFEs have played an overlooked role in these challenges as it seems its contributions to poverty reduction have taken center stage in international discourses, with its negative impacts on the environment being relegated to the backseat when considering the causes of environmental degradation. Attempts to effectively manage the activities of SMFEs have witnessed the emergence of a lot of challenges that threaten the very sustainability the globe yearns for. Some reasons point to the source of the challenges that have plagued these efforts, some of which are highlighted below.First is the lack of resources to recruit and train the needed personnel to constantly monitor the activities of these SMFEs during the harvesting of raw materials. This makes it easier for them to enter restricted forest areas without the necessary documentation and proceed to harvest more than they are required to at any given time. Secondly, it is difficult to track their activities because many SMFEs currently, do not register their businesses as required by law.A typical example is the use of unapproved trails or routes and the use of inappropriate harvesting techniques such as burning. This leads to the destruction of various lifeforms that are critical to the regenerative capabilities of the forests28. The study also found that the supervision of the activities of SMFEs is very poor as only 12% of SMFEs had their activities supervised on certain occasions. This buttresses the assertion by Acheampong et al.29 who posited that the lack of supervision is a major issue that needs to be vigorously addressed if we need to achieve forest sustainability in developing countries.There is a need to educate SMFEs on the laws and regulations governing the use of forest resources. It was revealed that only 16% of the respondents have some knowledge of the regulations governing the harvest and use of both woody and non-woody forest resources. This knowledge gap is being exploited by SMFEs as an excuse for not doing what is expected of them. However, a study found that 69% of respondents claimed to have good knowledge of the regulations governing their activities14. This can be attributed to self-learning or the action of the supervising authorities who for one reason or another other can perform their mandate of educating the SMFEs. There is a need to properly equip the supervising agencies to carry out this mandate.The research, therefore, cites the non-registering of SMFEs as an underlying cause of the flouting of these regulations and laws. The research also suggests that some form of training can be done at the point of registering even before the certification is done. As observed in the area of training, there is not enough emphasis on the need to train SMFEs in sustainability issues in terms of harvesting raw materials. It was noted that the majority (67%) of SMFEs (Table 8) have no training on how to harvest, process, and adequately market their products to ensure maximum profits while sustaining the resources for future harvests. There is a need to institute training and capacity-building programs for SMFEs that will empower them to succeed and yet aim to ensure sustainable forest management.The role of sustainable forest management in climate change mitigationSustainable forest management (SFM) can play a significant role in climate change mitigation, as forests are an important sink for carbon dioxide and other greenhouse gases. By sequestering carbon in their biomass and soils, forests can help to remove carbon dioxide from the atmosphere, which can help to mitigate the impacts of climate change30.There are a number of ways in which SFM can support climate change mitigation, including through the conservation and expansion of forests, the sustainable management of forests, and the use of forest-based products and practices that reduce greenhouse gas emissions. Policymakers and stakeholders at local, national, and international levels are increasingly recognizing the role of forests in climate change mitigation, and there is growing interest in developing strategies and policies that support the use of forests for this purpose.However, there are challenges that impede the efficient leveraging of SFM for climate change mitigation and one of such challenges is the need to balance economic, social, and environmental considerations31. Forests provide a range of goods and services that are vital for human well-being and economic development, including timber, non-timber forest products, and ecosystem services such as carbon sequestration, water regulation, and habitat for wildlife32. However, these resources can be in high demand, and managing forests sustainably can be difficult, particularly in developing countries where there may be limited access to financial and technical resources33.Another challenge is the impact of external factors such as climate change on the health and productivity of forests34. Rising temperatures and changing weather patterns can affect the growth and survival of forests, and may also increase the risk of forest fires and pests35. Policymakers must consider the role of forests in mitigating and adapting to climate change, as well as the potential impacts on forest-dependent communities32.One way in which SMFEs can contribute to climate change mitigation is through the sustainable management of forests. By practicing sustainable forestry, SMFEs can help to maintain and enhance the carbon sequestration capacity of forests, which can help to remove carbon dioxide from the atmosphere and mitigate the impacts of climate change31. This can involve practices such as planting and reforestation, soil and water conservation, and the use of sustainable harvesting techniques32. However, this study revealed the majority of these SMFEs are unregistered and therefore not monitored. Meaning their activities cannot be regulated to ensure practices that promote climate change mitigation.SMFEs can also contribute to climate change mitigation by using forest-based products and practices that reduce greenhouse gas emissions. For example, the use of wood products as a substitute for fossil fuel-based products can help to reduce emissions, as wood products sequester carbon over their lifetime and do not release it into the atmosphere when they are used34. In addition, the use of biomass energy in place of fossil fuels can help to reduce emissions, provided that the biomass is sourced sustainably and the emissions associated with its transportation and use are accounted for35.Another way in which SMFEs can contribute to climate change mitigation is through the development of innovative solutions and technologies that support sustainable forestry practices and reduce greenhouse gas emissions. This could include the use of precision forestry techniques, which use advanced technology to improve the efficiency and sustainability of forestry operations34. It could also involve the development and commercialization of new forest-based products or practices that have a lower carbon footprint32.Policies can have a significant impact on the way in which forests are managed for climate change mitigation31. For example, policies that promote sustainable forestry practices, such as the use of certification schemes or incentive programs, can help to ensure that forests are managed in a way that meets the needs of current and future generations33. On the other hand, policies that do not adequately consider the needs and interests of all stakeholders, or that do not provide sufficient support for sustainable forestry practices, may have negative impacts on the ability of forests to contribute to climate change mitigation34.Overall, addressing the inter-challenges of SFM for climate change mitigation and the impact of policies is an important part of ensuring the sustainability and long-term viability of forests as a tool for mitigating climate change.Development of SMFEs within the forest-based economy of Ghana through policyDespite the global consensus on the sustainability of forest resources and their utmost importance regarding the sustenance of present and future generations, the situation remains unclear at the field level36. The application of criteria and indicators of sustainability provides support for a small but crucial clarification on achieving sustainable forest management (SFM). A meaningful basis for assessing SFM at operational levels will require clarification together with management prescriptions and performance standards while providing linkage to voluntary timber certification.Currently, many environment-based non-governmental organizations (ENGOs) like Global Footprint Network and Fauna & Flora International who are concerned about natural resource exploitation, are convinced by the international debate on criteria and indicators that timber harvesting and ecosystem services of the forests can be sustained37. Stakeholders of the forestry franchise agree that environmental conservation can be accommodated through a necessary and reasonable modification and adaptation of forest-harvesting practices. Therefore, multi-resource forest management as a new paradigm replaces the indigenous sustained-yield management approach that bases on growth-harvest equilibrium using policy as a vehicle38.Food and Agriculture Organization (FAO) is assisting countries through policy advice, technical assistance, capacity building, workshop, and hands-on training, to overcome the challenges of sustainable forest management39. The assistance is provided through the assessment of forest resources and the elements of SFM, as well as the monitoring of progress toward it. FAO also identifies, tests, and modern scientific SMF approaches and techniques to address climate change mitigation and adaptation challenges such as increasing demand for wood and non-wood forest products and services, pest, and diseases.The views held by the Forestry Commission and National Board for Small Scale Industries (NBSSI) during interviews are in line with the suggestions and actions by the World Bank and FOA that involve training and other support systems for managers of forest resources in tropical countries like Ghana that depends heavily on its natural forests. Despite the availability of some of the avenues needed to execute these strategies, the non-compliance by SMFEs makes it difficult for these targets to be met. The general thought is that, if all relevant authorities and stakeholders perform their roles effectively, the current challenges of maximizing the contributions of SMFEs to development and sustainable forest management can be realized.The impact of forest policies is evident in countries like Gabon, a country rich in forest resources, which regards forests as a critical economic resource. World Bank-supported reforms have helped make concessions awarding procedures more competitive and transparent40. Forest taxation recovery has been bolstered, with tax collection rates increasing from 40 to 80% between 2005 and 2010. Sustainable forest management is presently practiced in around 85% of productive forest areas and as a result of these reforms, the forestry sector’s contribution to Gabon’s GDP increased from 2.5% in 2004 to 4.7% in 200940.Support for small and medium forest-based firms raised actual cash income among forest user groups by 53% in India’s Andhra Pradesh throughout the project duration. Seasonal outmigration decreased by 23%, and the quality of thick forest cover in these places improved40,41.Ghana has made significant progress toward sustainable management of its forest resources via the adoption of different forest regulations like the Forest and Wildlife Policy of 1994, Timber Resources Management Act, of 2002, etc. The problem with most of the country’s forest resource policies is the lack of attention paid to the human component; the emphasis is on sustainable timber extraction, even if it is destructive to the livelihoods of forest-dependent populations. Forest policies have historically been determined by successive administrations’ economic interests, which essentially focused on the exploitation of wood resources for income production. This has been a significant impediment to the creation and development of non-timber forest products which the majority of SMFEs depend on in Ghana. This has allowed the number SMFEs rapidly increase due to the lack of coverage by forest policies42. The policy interventions in Gabon and India have yielded results that can provide the foundation needed for Ghana to formulate its policies for the development of SMFEs in a way that does not threaten sustainable forest management. More