Philippa Kaur

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

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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

Study areaThe study area is located in Lintong District, Xi’an City, Shaanxi Province, China (34° 21′ 59.94″, 109° 12′ 51.012″) (Meteorologists, 2020b). The study area is located in northwestern China (Fig. 1), which is a Warm temperate semi-humid continental climate with distinct cold, warm, dry and wet seasons. Winter is cold, windy, foggy, and with little rain or snow. Spring is warm, dry, windy, and variable. The summer is hot and rainy, with prominent droughts and thunderstorms, and high wind. Autumn is cool, the temperature drops rapidly and autumn showers are obvious. The annual average temperature is 13.0–13.7 °C, the coldest January average temperature is −1.2–0 °C, the hottest July average temperature is 26.3–26.6 °C, the annual extreme minimum temperature is −21.2 °C, Lantian December 28, 1991, the annual extreme maximum temperature is 43.4 °C, Chang’an June 19, 1966. Annual precipitation is 522.4–719.5 mm, increasing from north to south. July and September are the two obvious peak precipitation months. The annual sunshine hours range from 1646.1 to 2114.9 h. The dominant wind direction varies from place to place, with the northeast wind in Xi’an, west wind in Zhouzhi and Huxian, east-northeast wind in Gaoling and Lintong, southeast wind in Chang’an, and northwest wind in Lantian. Meteorological disasters include drought, continuous rain, heavy rain, flooding, urban flooding, hail, gale, dry hot wind, high temperature, lightning, sand and dust, fog, haze, cold wave, and low-temperature freeze.
Figure 1Location of the field of study (The satellite imagery supporting this study was obtained using Baidu Maps (Android version—16.4.0.1195). The URL is (https://map.baidu.com/@14256795.568410998,5210675.606268121,8.67z.).Full size imageWheat (XiNong 805) was planted on September 24, 2019 and matured for harvest on May 28, 2020 (We warrant that we have the right to collect and manage wheat (XiNong 805). In addition, the study is in compliance with relevant institutional, national, and international guidelines.). Among the six strategies in the experiment (Table 1), we focused on strategies 1 and 4, fixed irrigation dates optimization and fixed fertilizer application dates optimization. Based on the custom of the study area, three days of diffuse irrigation were selected for Strategy 1. Three days of fertilization of the urea and three days of irrigation were selected for Strategy 4. The best practice for Strategy 1 was total irrigation of 201 mm for the total season and a total of 7388 kg/ha of wheat was obtained for this simulation, while the best practice for Strategy 4 was total irrigation of 197 mm for the total season and a total fertilizer application of 282 kg/ha for the total season. A total of 7894 kg/ha of wheat was obtained for this simulation.Table 1 Details of the 6 strategies of the experimental setup.Full size tableDSSAT modelDSSAT, one of the most widely used crop growth models, is an integrated computer system developed by the University of Hawaii under the authority of the U.S. Agency for International Development (USAID). It aims to aggregate various crop models and standardize the format of model input and output variables to facilitate the diffusion and application of models7, thereby accelerating the diffusion of agricultural technology and providing decision making and countermeasures for the rational and efficient use of natural resources in developing countries.
The DSSAT 4.5 model integrates all crop models into the simulation pathway-based CSM (Cropping System Model) farming system model, which uses a set of simulated soil moisture, nitrogen, and carbon dynamics codes, while crop growth and development are stimulated through the CERES37,38, CROPGRO39, CROPSIM, and SUBSOR modules. DSSAT is applicable to single sites or same type zones and can be extrapolated to the regional level through Geographic Information System (GIS).DSSAT–CSM simulates the growth process of crops grown on a uniform land area under prescribed or simulated management40, and the changes in soil water, carbon and nitrogen with under tillage systems. The DSSAT model is a decision support system supported by crop simulation models, which, in addition to data support, provides methods for calculating and solving problems, and provides decision-maker with the results of their decisions. It also provides scientific decisions for farmers to provide different cultivation management measures (e.g., proper fertilization and irrigation for crops) in different climatic years.Inputs and outputs of the modelThe DSSAT model has four main user-editable input files and various output files. The input files include crop management7,41, soil, weather, and cultivar parameter files; the output files include three types: (1) output files, (2) seasonal output files, and (3) diagnostic and management files.Crop management data: Crop management data provides basic information about crop growth. Detailed and accurate parameter provision is the basis for improving the accuracy of model simulation. Crop management parameters include crop variety, soil type, meteorological name, previous season crop, sowing period, sowing density, sowing depth, irrigation amount and time, fertilizer application amount and time, the initial condition of the soil, pest management, tillage frequency and method, etc. Some of these parameters are not easily available in field experiments and can be obtained from other test sites or from existing documentation. On the other hand, if there are missing values in the model, it will increase the simulation error of the model (this situation is hard to avoid). Therefore, in this study, the parameters were selected based on the principle of being both detailed and easily available.Soil data Soil data contains various parameters of the soil section plane, including soil color, soil slope, soil capacity, organic carbon, soil nitrogen content, drainage properties, the proportion of clay, particles, and stones in the soil. Similar to the governing documents, the more complete the parameters the smaller the error value of the simulation. The various physical and chemical properties of the soil for this study were obtained from the China Soil Database at the time of the study. The various physical and chemical properties of the soil for this study were obtained from the China Soil Database.Weather data The DSSAT model uses daily weather data as weather input data for the model. The model requires a minimum of four daily weather data in order to accurately simulate the water cycle in soil plants (Fig. 2). These are:(1) daily solar radiation energy (MJM); (2) daily maximum temperature (°C); (3) daily minimum temperature (°C); and (4) daily precipitation (mm). Weather data were obtained from the China Meteorological Administration. Weather data were obtained from the China Meteorological Administration.Figure 2Precipitation and maximum and minimum temperatures during 2019–2020.Full size imageModel calibration Adjusting the cultivar parameter is very important to accurately simulate the local growing environment. In this experiment, we collected field data for 2019 and 2020, and adjusted the parameters in the cultivar parameter files by trial-and-error method to make the simulation process more closely match the actual local crop growth process.Multi-objective optimization algorithmMulti-objective optimization techniques have been successfully applied in many real-world problems. In general42,43,44, MOPs produce a set of optimal solutions that together represent a trade-off between conflicting objectives, and such solutions are called Pareto optimal solutions (PS). These PS cannot make any solution better without compromising the other solutions. Therefore, when solving multi-objective problems, more PS are needed to find. Some MOPs aim to find all PS or at least a representative subset of them.A multi-objective optimization problem can be stated as follows:$$mathrm{min }Fleft(xright)={({f}_{1}left(xright),dots ,{f}_{k}(x))}^{T}$$
(1)
$$mathrm{subject;to};xin Omega$$
(2)
where (Omega) is the decision space,(F:Omega to {R}^{k}) consists of (k) real-value objective functions and ({R}^{k}) is called the objective space. The attainable objective set is defined as the set ({F(x)in Omega }).NSGA-II optimizerWe use non-dominated sorting genetic algorithm (NSGA-II) for Multiobjective optimization in R language. The NSGA-II algorithm is a classical multi-objective evolutionary algorithm with remarkable results in solving 2-objective and 3-objective problems45. It maintains the convergence speed and diversity of solutions by fast non-dominated sorting and crowding distance, selects the next population by elite selection strategy.Objective functionThe multi-objective optimization problem varies one or more variables to maximize or minimize two or more objective problems. In the case of crop production, where decision-makers change irrigation and fertilizer application to maximize benefits, this study focuses on when to apply irrigation or fertilizer on the field and how much irrigation or nitrogen fertilizer to apply.There are many crop models available that can be used as optimization objective functions, and DSSAT is definitely the best choice because it is easy to use and well-proven36. The user runs the model by entering defined soil, weather, variety, and crop management files, which are fed into the core of the model, the Crop Simulation Model (CSM). The model simulates the growth, development, and yield of crops grown on a uniform land area under management, as well as changes in soil water, carbon, and nitrogen over time under cropping systems. The CSM itself is a highly modular model system consisting of a number of sub-modules. Researchers have validated the output of these sub-modules as a whole under various crops, climate, and soil conditions.Using DSSAT, it is easy to design a set of objective functions and optimize them, as in our case.$$mathrm{Max}:Y=mathrm{DSSAT}left.left( {i}_{a0},dots ,{i}_{mathrm{aj}},{f}_{mathrm{a}0},dots ,{f}_{mathrm{ad}},{D}_{i}right.right)$$
(3)
$$mathrm{Min}:I=sum_{n=0}^{j}{i}_{an}$$
(4)
$$mathrm{Min}:F=sum_{m=0}^{d}{f}_{am}$$
(5)
where (Y) is yield,(I) is the total amount of irrigation, (F) is the total amount of nitrogen application, ({i}_{an}) is the amount of irrigation at one time, ({f}_{am}) is the amount of nitrogen applied at one time, (j) is a number of applications of irrigation, and (d) is a number of nitrogen applications. ({D}_{i}) is a random date combination of irrigation time and fertilizer application time.All other variables (e.g., climate, soil, location, crop variety) are kept constant during the optimization process. The irrigation unit is mm and the nitrogen application unit is kg/ha, the irrigation and nitrogen application amounts are positive integers by default (integer arithmetic reduces the program running time).Data-driven evolutionary algorithmsIn general, the key to DDEAs is to reduce the required FEs and assist evolution through data. The data is generally utilized through surrogate model. The use of suitable surrogate model can be used in place of real FEs46. Thus, DDEAs have more advantages over EAs in solving expensive problems.In terms of algorithmic framework, DDEAs contain two parts: surrogate model management (SMM) and evolutionary optimization part (EOP)47,48. The SMM part is used in order to obtain better approximations, while EOPs will use surrogate models in EAs to assist evolution. DDEAs can be divided into two types: online DDEAs and offline DDEAs23. Online DDEAs can be evaluated by real FEs with more new data. This new information can provide SMM with more information and construct a more accurate surrogate model49. Since DSSAT can obtain new data through FEs during the EOP process, the method used in this paper is online DDEAs. In contrast, offline DDEAs can only drive evolution through historical data.Radial Basis Function (RBF) network is a single hidden layer feedforward neural network that uses a radial basis function as the activation function for the hidden layer neurons, while the output layer is a linear combination of the outputs of the hidden layer neurons. RBF was used to approximate each objective function. According to the investigation of multi-objective optimization problems with high computational cost, radial basis functions are often used as the surrogate model, mainly because RBF networks can approximate arbitrary nonlinear functions with arbitrary accuracy and have global approximation capability, which fundamentally solves the local optimum problem of BP networks, and the topology is compact, the structural parameters can be learned separately, and the convergence speed is fast.In this paper, a new data-driven approach is proposed and place it in the lower-level optimization of the framework. RBF is utilized as the surrogate model and NSGA-II as the optimizer. Details are described in Algorithm 1.Data-driven method details
In step 1, the initial parent population is generated by randomly selecting points and the size is (N). In step 2, we run DSSAT (N) times to determine the objective function values of the (N) initial population solutions. Next, the algorithm then loops through the generations. At the beginning of each loop, surrogate models, which one objective train one surrogate and denoted by ({s}_{t}^{left({f}_{1}right)}) , were trained by the already obtained objective function values (step 3.1). The trial offspring ({P}_{i}^{^{prime}}left(tright)={ {x}_{1}^{^{prime}}left(tright),dots ,{x}_{u}^{^{prime}}left(tright)}) are generated by SBX and PM (step 3.2), then the trained surrogate model is used to predict the objective function values of trial offspring (step 3.3). The predicted objective function values are sorting by Pareto non-dominated and crowding distance (step 3.4), then (r) offspring (Q_{i} left( t right) = left{ {x^{primeprime}_{1} left( t right), ldots ,x^{primeprime}_{r} left( t right)} right}) are selected from the trial offspring (step 3.5).The offspring are evaluated by the DSSAT (step 3.6), and after combining the parent population and offspring population (step 3.7), the new parent population are selected by Pareto non-dominated and crowding distance sorting (step 3.8).Maximum extension distanceMED guides a small number of individuals to approximate the entire PF. MED is defined as follow:$$mathrm{MED}left({P}_{t}^{left(qright)}right)=mathrm{ND}left({P}_{t}^{left(qright)}right)times mathrm{TD}left({P}_{t}^{left(qright)}right)$$
(6)
where$$mathrm{ND}left({P}_{t}^{left(qright)}right)=underset{z,qne z}{mathrm{min}}sum_{m=1}^{M}left|{f}_{m}^{left(qright)}-{f}_{m}^{left(zright)}right|$$$$mathrm{TD}left({P}_{t}^{left(qright)}right)=sum_{z=1}^{P}sum_{m=1}^{M}left|{f}_{m}^{left(qright)}-{f}_{m}^{left(zright)}right|$$({P}_{t}^{left(qright)}) is the qth individual in population Pt at the tth generation. (mathrm{ND}left({P}_{t}^{left(qright)}right)) calculates the minimum distance between ({P}_{t}^{left(qright)}) and ({P}_{t}^{left(zright)}). The larger (mathrm{ND}left({P}_{t}^{left(qright)}right)) value means a better individual diversity. (mathrm{TD}left({P}_{t}^{left(qright)}right)) calculates the summation of distance between ({P}_{t}^{left(qright)}) and ({P}_{t}^{left(zright)}). The larger (mathrm{TD}left({P}_{t}^{left(qright)}right)) value means that the solution ({P}_{t}^{left(qright)}) has moved away from other individuals. A larger MED value means that an individual extends the overall boundary and an individual acquires better diversity.Modeling processTo maximize crop yield and optimize the use efficiency of water and fertilizer in a given environment, BSBOP framework is proposed. Crop growth is simulated by DSSAT, the data-driven approach reduces the runtime of the overall framework while finding optimal management strategies. The overall framework includes four main parts: upper-level screening, upper-level optimization, lower-level optimization and lower-level screening (Fig. 3).Figure 3Proposed integrated bi-level screening, bi-level optimization and DSSAT framework.Full size imageUpper-level screening The weather file in DSSAT was loaded by R language. The data are pre-processed with precipitation and solar radiation information to narrow down the date range for irrigation and fertilizer application. In other words, the date ranges for selecting irrigation and fertilization are restricted by the ULS.Upper-level optimization Generating random combinations of dates by the Latin hypercube sampling method (LHS). The upper-level screening starts with referencing the two variables (number of irrigation and nutrient application events). LHS uses these variables to generate a series of uniformly distributed random day combinations. For example, date combinations generated by the LHS could be May 15, July 18 and August 1 for irrigation and May 30, June 30 and July 18 for nutrient application. From the series of uniformly distributed random day combinations, one will be selected and incorporated into the lower-level optimization.Lower-level optimization The agricultural management strategy is optimized by the online data-driven approach proposed in Algorithm 1. Assuming three irrigation and three nitrogen application events are given, these events will be incorporated into the LOP, which consists of the RBF and NSGA-II. The population size of this paper is 105. The number of iterations varies according to the different strategies, and the objective function values are calculated by DSSAT. The main idea of applying Evolutionary multi-objective algorithms(EMO) and RBF to DSSAT is to generate a large number of trial offspring by traditional Simulated Binary Crossover (SBX) and Polynomial Mutation (PM), and then evaluate them using the trained surrogate model50. The objective values of the evaluation were then ranked by Pareto non-dominated and crowding distance, and the top 105 individuals were selected from a large number of trial offspring, after which a small number of individuals out of 105 were selected by Maximum Extension Distance (MED) for real function evaluation, and then combine the parents and offspring to select the next generation of parents by Pareto non-dominated and crowding distance sorting. Furthermore, in the numerical experiments, to ensure the superiority of the algorithm and reduce the experimental complexity, we use a relatively simple radial basis function (RBF) surrogate. The NSGA-II algorithm can be used for both bi-objective and tri-objective problems, so it can optimize the system by starting with the most critical objective and then adding additional objectives. For each solution in the population, the objective functions (1: maximize yield, 2: minimize irrigation application, 3: minimize nitrogen fertilizer application) will be evaluated by invoking the DSSAT model for these dates and the amount of fertilizer irrigation applied. Populations will be tested against the termination criteria (maximum number of iterations allowed). If the termination criteria are not satisfied, the population evolves and is re-evaluated again. The process is repeated until the termination criterion is satisfied and then the local Pareto front of the selected day combination is stored. After each iteration of the UOP, the new local Pareto is combined with the global Pareto frontier. In the next step, if there are any remaining day combinations, the above process is repeated for each new day combination until all generated random day combinations have been processed.Lower-level screening Firstly, the K-means method is used to screen the global Pareto solutions with higher yield. Then, secondary screening takes economic efficiency as the objective and optimizes it by Differential Evolution (DE) algorithm. Finally, the locally appropriate solution is intelligently selected.Optimization strategies and configurationDue to the complexity of the problem, a BSBOP framework was proposed in this study. Due to a large number of variables behind irrigation and fertilization, traversal date for optimization appears to be particularly difficult and time-consuming, assuming that only irrigation is optimized for 120 days of the growth cycle and the decision-maker has 0-150 mm of water per day, then there are ({151}^{120}) different solutions. If both irrigation and fertilization are considered, then there are ({151}^{120}cdot {151}^{120}) different solutions. Therefore, this study tries to reduce the number of variables while minimizing the running time of the algorithm.Here we hypothesize that more precision and effective agricultural management can be implemented through the proposed framework. Not only can crop yields be increased, but also irrigation application and fertilizer application can be reduced, while the solutions obtained have important guidance for decision-makers: such as the selection of irrigation and fertilizer application dates during the growing season of the crop, the selection of irrigation and fertilizer application amounts, and the relationship between economic benefits and application costs. To test this hypothesis, different optimization strategies were developed and evaluated (Table 1). Each optimization strategy was aimed at maximizing yield while minimizing resource wastage.The various strategies are listed below (Table 1). Strategy 1—Fixed irrigation dates: Keeping the number of irrigation days and all parameters constant, only the amount of irrigation on each date is changed, trying to reduce the amount of irrigation as much as possible, make it easy to compare the results with best practices. Strategy 2—Optimal irrigation dates: Traverse through the irrigation dates to optimize irrigation, and try to find a better combination of irrigation dates (optimal dates) and better amount of irrigation over the wheat growth cycle. Strategy 3—Optimal irrigation dates based on surrogate model: RBF is added to Strategy 2, which makes it possible to reduce lots of time. Strategy 4—Fixed fertilizer application date: Using the optimal irrigation date found in Strategy 2 while keeping the number of days of fertilization and all other parameters constant, irrigation and fertilization are optimized in an attempt to minimize the amount of irrigation and fertilizer applied. Strategy 5—Optimal fertilizer application date: while ensuring the optimal irrigation date, traverse the fertilizer application date for optimization, trying to find out the potential yield of the crop. Strategy 6—Optimal fertilizer application date based on surrogate model: RBF is introduced based on Strategy 5. The time consumption was reduced.The stopping criterion in this study is when the optimization results converge visually. The algorithm population size was set to 105, and the generation of offspring used traditional polynomial Mutation. The number of hidden layers of the surrogate model is equal to the dimension of the decision variables, the learning rate is 0.01, the Gaussian kernel function is chosen as the activation function of the hidden layer in the RBF network. The neurons centers are generated by the K-means clustering method. The width parameter of the function is generated by calculating the variance of each cluster. The optimization weight parameters are selected by the recursive least square method. This is because the use of the least square method is likely to encounter situations where matrix inversion is troublesome. Therefore, recursive least squares (RLS) is often used to give a recursive form of the matrix in which the inverse needs to be found, making it computationally easier. More

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