In this paper, we have fuzzified the nonlinear mathematical model given by Devi and Mishra [1] to compare the dynamics of carbon dioxide (CO2) in crisp and fuzzy environments. Reserved forestry biomass is necessary to control the dynamics of CO2. Devi and Mishra [1] have studied the dynamics of CO2 under a system consisting of reserved forestry biomass, unreserved forestry biomass, and human population in a crisp environment. We have analyzed the dynamics of CO2 under the system consisting of reserved forestry biomass, unreserved forestry biomass, and human population in crisp and fuzzy environments. Conditions for boundedness of solutions, existence, and stabilities of equilibrium points are discussed for the fuzzified model system. We have performed numerical simulations to validate our analytical findings and to see the comparison in the dynamics of CO2 between crisp and fuzzy environments. In this study, many notable differences were found in the dynamics of CO2 in crisp and fuzzy environments. (c) 2026 L&H Scientific Publishing, LLC. All rights reserved.

Tourism significantly benefits communities near popular destinations by generating employment, enhancing local economies, and preserving cultural traditions. However, the influx of tourists and associated tourist activities can harm local forest biomass through the introduction of precursors and toxicants. To explore and mitigate these effects, a nonlinear mathematical model is developed to study the dynamics of forest biomass in hill regions and assess the effectiveness of external interventions and awareness programs. The model incorporates interactions among forest biomass, tourist population, precursors, toxicants, awareness programs, and external efforts. Analytical results establish the conditions for boundedness and persistence of the solutions, existence, and uniqueness of equilibrium points, and their stabilities. Sensitivity analysis reveals the model’s robustness to parameter variations, and optimal control strategies are introduced to minimize toxicant levels. Numerical simulations confirm the effectiveness of these strategies and highlight the critical role of awareness programs and external interventions in sustaining forest biomass. The findings underscore the urgent need to regulate tourist-induced environmental stress in hilly areas through awareness programs and strategic efforts.

Age structure of trees or plants is a crucial phenomenon because in a forest stand, the capture and storage of carbon dioxide (CO_2) depend on their age and number. This study presents a nonlinear mathematical model to explore the consequences of the capacity of plants/trees to absorb atmospheric CO_2 . For the formulation of model, we divide the whole plant population into four groups, namely: small plants (like, herbs and shrubs), early age tree (sapling), middle age tree (mature) and old age tree. To analyze the qualitative behavior of model, we focused on to investigate the region of attraction and permanence of solutions, and carried out the conditions to ensure the existence of all equilibria and their stabilities. Also, it is noticed that for a sudden depletion rate exceeding a critical threshold of small plants may trigger a transcritical bifurcation resulting the disappearance of small plants. To corroborate the analytical results of our mathematical model, numerical simulations are conducted. Effects of crucial parameters are compared for each category of plants/trees to absorb CO_2 which depict the importance of categorization.

Modelling of predator–prey interactions has vast history in ecological theory and these models have been developed in large scale since the initial Lotka–Volterra model in order to explain the processes of predation, cooperation and diffusion more realistically. Introducing Allee effect and diffusion in predator–prey models are very important facet of these evolutions, where per capita growth rate of species increases with increase in population density and species move from region with low population density to region with high population density. In this paper, dynamics of a predator–prey model considering strong Allee effect, diffusion and Holling type-I function response is studied. Existence and stability of equilibrium points are discussed. Analysis of the model reveals that by applying some conditions on the parameters, diffusion can induce Turing instability. Furthermore, conditions for the existence of Hopf bifurcation are obtained for both the ODE and PDE models. Bifurcation direction and the stability of the bifurcating periodic solution are determined by center manifold theorem and the normal form theory. Numerical simulations are performed to support our mathematical findings. By this study, we see that if Allee threshold crosses a critical value, the interior equilibrium point losses its stability. Therefore, system undergoes Hopf bifurcation and periodic solution arises. Here, we note that Allee effect plays an important role in the dynamics of predator–prey model. We also observe that diffusion destabilized the model which was stable in the absence of diffusion.

In this paper, we have proposed a nonlinear mathematical model to study the dynamics of atmospheric carbon dioxide $({\rm{CO}}{{\kern 1pt} _{\rm{2}}})$(CO2) and atmospheric temperature due to energy sectors. Energy sector is a major contributor of atmospheric carbon dioxide. We have studied the effects of energy consumption and human population on ${\rm{CO}}{{\kern 1pt} _{\rm{2}}}$CO2 level and temperature. Burning of fossil fuels results in an increase of ${\rm{CO}}{{\kern 1pt} _{\rm{2}}}$CO2 level and temperature. To curb the ${\rm{CO}}{{\kern 1pt} _{\rm{2}}}$CO2 level and temperature, efficient mitigation options are required. Mitigation options, which reduce emission rate of ${\rm{CO}}{{\kern 1pt} _{\rm{2}}}$CO2, energy consumption rate and rate of increase in temperature, are considered in this model. By taking the efficiencies of mitigation options as control variables, the optimality system is derived. Numerical simulations are carried out to validate our analytical findings. The optimal profiles of control variables are plotted for different values of emission rate of ${\rm{CO}}{{\kern 1pt} _{\rm{2}}}$CO2, energy consumption rate and rate of increase in temperature. Maximum efficiencies of mitigation options are also plotted against to reduce emission rate of ${\rm{CO}}{{\kern 1pt} _{\rm{2}}}$CO2, energy consumption rate and rate of increase in temperature. It is found that mitigation cost of implementation of mitigation options can be minimized by implementing more efficient mitigation options.

This paper introduces and examines a nonlinear mathematical model aimed to elucidate the interplay among soil fertility, earthworm population, and inorganic fertilizers concerning agricultural crop productivity. Within the modeling framework, we assume logistic growth of crops, influenced by soil fertility, earthworm population, and inorganic fertilizers. The impact of earthworms on crop yield is explored using a Holling type-II functional response. Furthermore, the use of inorganic fertilizers is presumed to adversely affect both soil fertility and earthworm population. Moreover, we hypothesize a direct influence of earthworm population on soil fertility. We carry out conditions for the boundedness of solutions, existence, uniqueness, and both local and global stability of equilibrium points. Numerical simulations are conducted to validate the analytical conditions. Optimal control strategies are employed to minimize the use of inorganic fertilizers in crop field. Graphical comparisons between models, with and without optimal control, reveal that the latter leads to a slight increase in crop production and improvements in both earthworm population and soil fertility.

To understand the impacts of efforts applied to increase the density of forestry biomass on the dynamics of forestry biomass, concentration of greenhouse gases and elevated temperature a nonlinear mathematical model is proposed and analysed. The mathematical model involves four dynamical variables namely: the density of forestry biomass, efforts applied to increase the density of forestry biomass, concentration of greenhouse gases and elevated environmental temperature. Since, there is always a time lag between implementation of efforts and its outcome, therefore we extend our model by introducing time delay in efforts. It is found that as delay in efforts crosses a critical value, the delay model loses its stability and undergoes Hopf bifurcation. The stability and direction of Hopf bifurcation are analysed using normal form method and central manifold theory. Numerical simulations are performed to verify and validate our analytical results. It is observed that if efforts are implemented for appropriate time, the density of forestry biomass can be conserved but implementation of efforts with longer time delay has destabilizing effect on the system. Increasing rate of atmospheric temperature due to greenhouse gases, decreases the density of forestry biomass but this decrease can also be maintained by implementation of efforts. Therefore, implementation of efforts for sufficient period of time, plays a very vital role in increasing the density of forestry biomass and decreasing the concentration of greenhouse gases and elevated temperature.

In this paper, we have formulated and analysed a mathematical model to investigate the impacts of lockdown on the dynamics of forestry biomass, wildlife species and pollution. For this purpose, we have considered a nonlinear system of four ordinary differential equations representing rates of change of the density of forestry biomass, the density of wildlife species, the concentration of pollutants and lockdown. Conditions for the existence, uniqueness and local stability of all equilibria along with the global stability of the interior equilibrium point are derived. Furthermore, conditions that influence the persistence of the system are obtained. By formulating an optimal control problem, the optimal strategies for minimizing the cost of implementation of lockdown as well as the concentration of pollutants are also studied. Numerical simulations are carried out to verify and validate our analytical findings. By this study, we have observed that implementation of lockdown for a sufficient period of time minimizes excessive harvesting of both forestry biomass and wildlife species and the concentration of pollutants in the environment. It is also found that lockdown policy is effective in the optimal control of atmospheric pollution. Therefore, lockdown plays a significant role in the dynamics of forestry biomass, wildlife species and control of pollution in the environment.

This paper deals with mathematical modelling and analysis of a SEIQR model to study the dynamics of COVID-19 considering delay in conversion of exposed population to the infected population. The model is analysed for local and global stability using Lyapunov method of stability followed by Hopf bifurcation analysis. Basic reproduction number is determined, and it is observed that local and global stability conditions are dependent on the number of secondary infections due to exposed as well as infected population. Our study reveals that asymptomatic cases due to exposed population play a vital role in increasing the COVID-19 infection among the population.

In this paper, an idea to impose environmental tax on the basis of greenhouse gas emission has been developed. A linear non homogeneous equation has been used to describe the dynamics of concentration of greenhouse gases which involves a term for environmental fax. A formula has been obtained to calculate amount of environmental tax to reduce the concentration of greenhouse gases to some extent in prescribed time. For fair imposition of environmental tax a compartmental model has been constructed here. Some numerical simulation also has been done to describe the results of the analysis.

In this paper, we have proposed and analyzed a mathematical model to conserve the endangered sparrows before it becoming close to extinct. Here, we have considered a nonlinear system consisting of five ordinary differential equations representing rates of change of sparrow’s population, human population, concentration of pesticides/insecticides, electromagnetic radiations emitted by mobile phones/mobile towers, and awareness campaigns, respectively with respect to time. System is analyzed in regard to boundedness, equilibria, stability, (both local as well as global) and persistence. Equilibrium value of each variable is ecologically interpreted with respect to awareness programmes. From our mathematical analysis and numerical simulations, we have pointed out that, if awareness campaigns are conducted regularly to aware the people about the eco-friendly importance of sparrows, so that they will minimize the indiscriminate use of pesticides/insecticides and mobile phones, then population of sparrows can be maintained at a suitable level and they will never disappear from our environment.

This paper deals with two well-known problems of ecology and environmental science, global warming and the threat to plant pollination, and their interrelationship. In this paper, a nonlinear mathematical model is proposed and analysed to study the dynamics of plant–pollinator interactions under the adverse effects of increasing environmental temperature due to the increasing concentration of greenhouse gases in the environment. The aim of this paper is also to provide some insights into the problem of global warming and to provide some solutions to the same. Analysis for existence, local stability, and global stability of the interior equilibrium point and uniform persistence of solutions of the system are carried out followed by numerical simulations. Results show that the huge emission of greenhouse gases into the atmosphere, not only reduces plant–pollinator interactions but it may also cause the extinction of the population of pollinators and plants. Results also suggest the immediate reduction of those types of greenhouse gases which cause warming of the environment with higher intensity. Our analysis points out the need for more researches to find out some techniques so that the cooling rate of the environment can be increased.

This paper deals with a non-linear mathematical model which is formulated by introducing the concept of reserved forestry biomass. The aim of the proposed mathematical model is to study a dynamical system consisting of the human population with reserved and unreserved forestry biomasses, and the increasing concentration of carbon dioxide in the atmosphere. Conditions for existence, uniqueness, and local stabilities of all possible equilibria are derived and ecologically explained. The conditions for global stability of the interior equilibrium point and uniform persistence of the system, are derived. Numerical simulations are also performed for further studies of the mathematical model. The analysis clearly shows that too much exploitation of forestry biomass results in their extinction and a huge increase in the concentration of atmospheric carbon dioxide. From the analysis, we have observed that the reserved forestry biomass is more effective in controlling the level of $$\hbox {CO}_2$$ in the atmosphere than the unreserved forestry biomass. Hence, by this study, we convey the message that besides the preservation of forestry biomasses, reserved forestry biomasses may play a significant role to save our environment too.

In this paper, a non-linear mathematical model is proposed and analysed to study the dynamics of plant-pollinator-pesticide interactions. Analysis of the existence of equilibrium points is done. Sufficient conditions for local as well as global stability have been derived. A threshold value of the energetic reward is obtained above which the dynamical system persists uniformly. Further, numerical simulations are performed to confirm the analytic results and to deduce some other important conclusions. It is shown that the plant and pollinator populations support the growth of each other and are mutually dependent when the energetic reward is high. It is observed that equilibrium values of plant and pollinator populations can be maintained for high energetic rewards even though pesticides are present, but at low values of energetic reward, plant population becomes extinct in presence of pesticides.

In this paper, three fuzzy nonlinear mathematical models, corresponding to crisp models described in Devi and Gupta (2018), are formulated to study the comparative behavior of carbon dioxide concerning vegetation biomass in crisp and fuzzy environments. Logistic growth and regrowth of vegetation biomass are considered in the first two models, and delay is incorporated in the regrowth of vegetation biomass in the third one. Conditions for boundedness and persistence of the solutions, the existence of equilibrium points, and the stability of equilibrium points are carried out for each model. We also derived the formula for the critical value of time delay. Further, we also analyzed the existence, stability, and direction of the Hopf bifurcation. Numerical simulations were performed to complement analytical findings and to see the comparison between crisp and fuzzy environments. In fuzzy environment, it is observed that increment in the minimum value of regrowth rate shifts the destabilizing value of delay forward while if the maximum value of regrowth rate decreases, variables will become unstable before the estimated time. The overall analysis demonstrates that the behavior of the models in crisp and fuzzy environments is quite different. Recommendations for resource managers An increasing level of CO2 is the major factor for greenhouse effect, therefore to reduce the destructive effects of global warming, implementation of policies related to mitigation of the atmospheric CO2 is of paramount importance. Vegetation biomass has considerable regrowth rate as compared with the rest of the plants. But, there is some time lag in the process of regrowth, therefore for the growth rate of vegetation biomass regrowth with delay must be considered to avoid the fluctuations in the results. In the real world, the biological parameters are not always fixed because they vary according to the environment. So, scientists should consider fuzzy parameters for more precise results.

In this research paper, we plan to study the relation among concentration of greenhouse gases, human population and environmental tax. For this purpose, a deterministic nonlinear mathematical model is proposed and analyzed in regard to the boundedness and persistence of its solutions, equilibria and their stabilities. Whenever, a policy is implemented, an automated delay comes into existence because outcomes of application of any policy always take some time to become visible. To see the effects of delay, analyses for the stability and direction of Hopf bifurcation for delay system are done. Critical value of the delay parameter is calculated theoretically and numerically, and then verified graphically. For verifications and descriptions of analytical findings, numerical simulations are performed. Graphical comparisons for different values of various parameters provides us some realistic and interesting results. Overall, model analysis shows that the increasing level of greenhouse gases can be controlled by applying environmental tax and imposing some restrictions on corruption. (C) 2019 Elsevier Ltd. All rights reserved.

Scientists commonly use linear or bilinear functional responses to relate density of biomass and concentration of CO2 which depicts that density of biomass always increases as concentration of CO2 increases. A study shows that although CO2 provides food security to the biomass but up to some extent i.e. even CO2 is necessary for the growth of biomass but much higher concentration has negative effect on the growth of biomass (like soybean, corn). Here we have formulated and analysed a mathematical model taking clue from above result to study the dependence of biomass on the concentration of CO2. Conditions of local and global stability of interior equilibrium point, boundedness as well as persistence have been derived. Bifurcation analysis also has been done for the harvesting parameter. Results of the model resembles with the realistic behaviour of biomass with respect to CO2.

To explore the effects of variation in the capability of plants to absorb atmospheric carbon dioxide (CO2), a nonlinear mathematical model is proposed here. The model is developed under the presumption that capability of plants to inhale/store CO2 varies plantwise. Primarily, whole plant population is divided into two categories for the analysis. Qualitative analysis for boundedness of solutions, existence, and stability of equilibrium points, along with permanence of the model system is carried out. The model analysis reveals that increasing the growth rate of plants, whose capability to absorb atmospheric CO2 is more, depleted CO2 more rapidly as compared to other ones, and if we increase harvesting rate coefficient, the concentration of CO2 increases accordingly. Numerical simulations are performed to support and interpret the analytical results. Finally, the plant population is categorized in n-parts and model is generalized to better understand the dynamics of the ecosystem.

This paper deals with a ratio-dependent predator-prey model with predator self limitation in which prey species use a constant refuge. Harvesting of only that population of prey species has been considered which is not using refuge. Conditions for local as well as global stability have been derived. Optimal harvesting policy has been discussed treating tax as control variable. By numerical simulations it is shown that the dynamic behaviour of the system is very much affected by the prey refuge parameter and the level of tax imposed. It is also shown that the self limitation parameter plays a very crucial role in maintaining the predator population at appropriate level when prey population increases or decreases.