Pub Date : 2022-05-05DOI: 10.1142/s0218339022500140
Shivam, Teekam Singh, Mukesh Kumar
This paper considers a diffusive prey–predator system with fear and group defense in the prey population. Also, we consider that the mortality of predators is linear and quadratic. By using local stability analysis, we get the prerequisite of Turing instability. Using comprehensive numerical computations, we get non-Turing pattern formation in the system with linear death of predator. Turing patterns are obtained for the system with the quadratic death of the predator. The modeling technique of multiple scale analysis is used to determine amplitude equations near the Turing bifurcation origin for the model with the predator’s quadratic mortality rate. The amplitude equations stability leads to various Turing patterns such as spots, stripes, and mixed. The result focuses on changing the mortality rate linear to quadratic of a predator in the prey–predator system. The derived results support us in a more immeasurable understanding of prey–predator interaction dynamics in the actual world.
{"title":"SPATIOTEMPORAL DYNAMICAL ANALYSIS OF A PREDATOR–PREY SYSTEM WITH FEAR AND GROUP DEFENSE IN PREY","authors":"Shivam, Teekam Singh, Mukesh Kumar","doi":"10.1142/s0218339022500140","DOIUrl":"https://doi.org/10.1142/s0218339022500140","url":null,"abstract":"This paper considers a diffusive prey–predator system with fear and group defense in the prey population. Also, we consider that the mortality of predators is linear and quadratic. By using local stability analysis, we get the prerequisite of Turing instability. Using comprehensive numerical computations, we get non-Turing pattern formation in the system with linear death of predator. Turing patterns are obtained for the system with the quadratic death of the predator. The modeling technique of multiple scale analysis is used to determine amplitude equations near the Turing bifurcation origin for the model with the predator’s quadratic mortality rate. The amplitude equations stability leads to various Turing patterns such as spots, stripes, and mixed. The result focuses on changing the mortality rate linear to quadratic of a predator in the prey–predator system. The derived results support us in a more immeasurable understanding of prey–predator interaction dynamics in the actual world.","PeriodicalId":54872,"journal":{"name":"Journal of Biological Systems","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42481016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-20DOI: 10.1142/s0218339022500115
Amel Ghouali, A. Moussaoui, P. Auger, Tri Nguyen Huu
In this paper, we propose to study a fishery model with variable price. We assume that the price evolves much faster than the rest of the system. Under certain assumptions, this makes it possible to consider the fishery system as a slow–fast system, on two time scales, and to study it with a reduced model of dimension two. Two main cases can occur. The first one which we called catastrophic equilibrium corresponds to over-exploitation leading to fish extinction and a booming price. The second case corresponds to a sustainable fishery equilibrium which is stable. The possible effects of the creation of marine protected areas (MPAs), sites where fishing is prohibited, on the fish stock and fishery are evaluated. We show that MPAs can have a positive effect on the restoration of depleted fish stocks by destabilizing the catastrophic equilibrium and keeping only one positive equilibrium which will be globally asymptotically stable. This problem is addressed by proposing a model with MPA for the fish dynamics. Fish are assumed to move between MPA and fishing area and are subject to harvesting through fishing. We show that to avoid the extinction of the stock and stabilize the fishery in the long term, it is necessary to define a fishing zone such that the ratio of its carrying capacity to its surface is small enough. We further show that with a judicious choice of the surface area of the MPA, it is possible to optimize the total capture.
{"title":"Optimal Placement of Marine Protected Areas to Avoid the Extinction of the Fish Stock","authors":"Amel Ghouali, A. Moussaoui, P. Auger, Tri Nguyen Huu","doi":"10.1142/s0218339022500115","DOIUrl":"https://doi.org/10.1142/s0218339022500115","url":null,"abstract":"In this paper, we propose to study a fishery model with variable price. We assume that the price evolves much faster than the rest of the system. Under certain assumptions, this makes it possible to consider the fishery system as a slow–fast system, on two time scales, and to study it with a reduced model of dimension two. Two main cases can occur. The first one which we called catastrophic equilibrium corresponds to over-exploitation leading to fish extinction and a booming price. The second case corresponds to a sustainable fishery equilibrium which is stable. The possible effects of the creation of marine protected areas (MPAs), sites where fishing is prohibited, on the fish stock and fishery are evaluated. We show that MPAs can have a positive effect on the restoration of depleted fish stocks by destabilizing the catastrophic equilibrium and keeping only one positive equilibrium which will be globally asymptotically stable. This problem is addressed by proposing a model with MPA for the fish dynamics. Fish are assumed to move between MPA and fishing area and are subject to harvesting through fishing. We show that to avoid the extinction of the stock and stabilize the fishery in the long term, it is necessary to define a fishing zone such that the ratio of its carrying capacity to its surface is small enough. We further show that with a judicious choice of the surface area of the MPA, it is possible to optimize the total capture.","PeriodicalId":54872,"journal":{"name":"Journal of Biological Systems","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44085275","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-20DOI: 10.1142/s0218339022500097
A. Mhlanga, T. V. Mupedza, T. M. Mazikana
Scabies is caused by sarcoptes scabiei var. hominis, which is also referred to as itch mice. The disease is transmitted through direct contact with an infected person, or from contact with infested bedding or clothing. In this paper, a mathematical model for the spread of scabies was proposed and analyzed. Sensitivity analysis of the model parameters was carried out. Optimal control theory was applied to our proposed model, with the controls representing treatment and vaccination. Our aim was to minimize cumulative infectious cases and susceptible individuals through treatment and vaccination, respectively. Pontryagin’s maximum principle was utilized to characterize the optimal levels of the two controls. The resulting optimality system was then solved numerically. The optimal control result was further highlighted by applying the results realized from the cost objective functional, the IAR, and the ICER.
{"title":"Optimal Control and Cost-Effective Analysis of a Scabies Model with Direct and Indirect Transmissions","authors":"A. Mhlanga, T. V. Mupedza, T. M. Mazikana","doi":"10.1142/s0218339022500097","DOIUrl":"https://doi.org/10.1142/s0218339022500097","url":null,"abstract":"Scabies is caused by sarcoptes scabiei var. hominis, which is also referred to as itch mice. The disease is transmitted through direct contact with an infected person, or from contact with infested bedding or clothing. In this paper, a mathematical model for the spread of scabies was proposed and analyzed. Sensitivity analysis of the model parameters was carried out. Optimal control theory was applied to our proposed model, with the controls representing treatment and vaccination. Our aim was to minimize cumulative infectious cases and susceptible individuals through treatment and vaccination, respectively. Pontryagin’s maximum principle was utilized to characterize the optimal levels of the two controls. The resulting optimality system was then solved numerically. The optimal control result was further highlighted by applying the results realized from the cost objective functional, the IAR, and the ICER.","PeriodicalId":54872,"journal":{"name":"Journal of Biological Systems","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48717344","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-12DOI: 10.1142/s0218339022500127
Zhibo Zhang, Sheng Li, Peng Si, Xuefang Li, Xiongxiong He
We develop a mathematical model of tumor-immune interactions, including six populations (tumor cells, CD8[Formula: see text]T cells, natural killer (NK) cells, dendritic cells, helper T cells, cytokine interleukin-12 (IL-12)) and three potential treatments (chemotherapy, Tumor-infiltrating lymphocyte (TIL) therapy and IL-12 therapy). We characterize the dynamics of our model without treatment through stability and sensitivity analysis, which provides a broad understanding of the long-term qualitative behavior. To find the best combination of the chemo-immunotherapy regimens to eliminate tumors, we formulate an optimal control problem with path constraints of total drug dose and solve it numerically with the optimal control software Pyomo. We also simulate the scenarios of traditional treatment protocols as a comparison and find that our optimal treatment strategies have a better therapeutic effect. In addition, numerical simulation results show that IL-12 therapy is a good adjunctive therapy and has a high potential for inhibiting a large tumor in combination with other therapy. In most cases, combination therapy is more effective than a single treatment.
{"title":"A TUMOR-IMMUNE MODEL WITH MIXED IMMUNOTHERAPY AND CHEMOTHERAPY: QUALITATIVE ANALYSIS AND OPTIMAL CONTROL","authors":"Zhibo Zhang, Sheng Li, Peng Si, Xuefang Li, Xiongxiong He","doi":"10.1142/s0218339022500127","DOIUrl":"https://doi.org/10.1142/s0218339022500127","url":null,"abstract":"We develop a mathematical model of tumor-immune interactions, including six populations (tumor cells, CD8[Formula: see text]T cells, natural killer (NK) cells, dendritic cells, helper T cells, cytokine interleukin-12 (IL-12)) and three potential treatments (chemotherapy, Tumor-infiltrating lymphocyte (TIL) therapy and IL-12 therapy). We characterize the dynamics of our model without treatment through stability and sensitivity analysis, which provides a broad understanding of the long-term qualitative behavior. To find the best combination of the chemo-immunotherapy regimens to eliminate tumors, we formulate an optimal control problem with path constraints of total drug dose and solve it numerically with the optimal control software Pyomo. We also simulate the scenarios of traditional treatment protocols as a comparison and find that our optimal treatment strategies have a better therapeutic effect. In addition, numerical simulation results show that IL-12 therapy is a good adjunctive therapy and has a high potential for inhibiting a large tumor in combination with other therapy. In most cases, combination therapy is more effective than a single treatment.","PeriodicalId":54872,"journal":{"name":"Journal of Biological Systems","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44640884","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-12DOI: 10.1142/s0218339022500103
Sasanka Shekhar Maity, P. Tiwari, Samares Pal
In this paper, we study a predator–prey system in which the prey population is infected from a parasite and the growth of susceptible prey is suppressed due to fear of predation. We consider that the predators have the ability to distinguish between the susceptible and infected prey items, and they avoid the infected ones to reduce fitness cost. The predators are assumed to die naturally and also due to intraspecific competition. The proposed model is analyzed mathematically for the feasibility and stability of the system’s equilibria. We also discuss the existence of Hopf bifurcation by taking the feeding preference of predators as a bifurcation parameter. We perform global sensitivity analysis to identify model parameters having significant impact on the density of predator population in the ecosystem. Our simulation results show the stabilizing role of selective feeding of predators whereas fear factor and disease prevalence induce limit cycle oscillations. Feeding more the predators with additional foods bring stability in the system by evacuating the persistent oscillations. To model the situation more realistically, we consider that the parameters representing the cost of fear and the feeding preference of predators vary with time. For the seasonally forced system, conditions are obtained for which the system has at least one positive periodic solution; global attractivity of the positive periodic solution is also discussed. Our seasonally forced model demonstrates the appearance of a unique periodic solution, higher periodic solutions and complex bursting patterns.
{"title":"AN ECOEPIDEMIC SEASONALLY FORCED MODEL FOR THE COMBINED EFFECTS OF FEAR, ADDITIONAL FOODS AND SELECTIVE PREDATION","authors":"Sasanka Shekhar Maity, P. Tiwari, Samares Pal","doi":"10.1142/s0218339022500103","DOIUrl":"https://doi.org/10.1142/s0218339022500103","url":null,"abstract":"In this paper, we study a predator–prey system in which the prey population is infected from a parasite and the growth of susceptible prey is suppressed due to fear of predation. We consider that the predators have the ability to distinguish between the susceptible and infected prey items, and they avoid the infected ones to reduce fitness cost. The predators are assumed to die naturally and also due to intraspecific competition. The proposed model is analyzed mathematically for the feasibility and stability of the system’s equilibria. We also discuss the existence of Hopf bifurcation by taking the feeding preference of predators as a bifurcation parameter. We perform global sensitivity analysis to identify model parameters having significant impact on the density of predator population in the ecosystem. Our simulation results show the stabilizing role of selective feeding of predators whereas fear factor and disease prevalence induce limit cycle oscillations. Feeding more the predators with additional foods bring stability in the system by evacuating the persistent oscillations. To model the situation more realistically, we consider that the parameters representing the cost of fear and the feeding preference of predators vary with time. For the seasonally forced system, conditions are obtained for which the system has at least one positive periodic solution; global attractivity of the positive periodic solution is also discussed. Our seasonally forced model demonstrates the appearance of a unique periodic solution, higher periodic solutions and complex bursting patterns.","PeriodicalId":54872,"journal":{"name":"Journal of Biological Systems","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43726292","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-24DOI: 10.1142/s0218339022500061
Yaru Guo, Shulin Sun
A stochastic non-autonomous one-prey two-predator model with Crowley–Martin functional response and impulses is proposed in this paper. First, by constructing the equivalent system without impulses, we investigate the existence and uniqueness of the global positive solution of the system. Second, by using Itô formula, strong law of large numbers and Chebyshev’s inequality, some sufficient conditions are established to ensure the extinction, non-persistence in the mean, persistence in the mean and stochastic permanence of the system. Third, we prove the system is globally attractive under some conditions. Finally, we choose different white noise intensities and impulsive parameters to illustrate the analytical results by numerical simulations.
{"title":"DYNAMIC BEHAVIOR OF A STOCHASTIC NON-AUTONOMOUS PREDATOR–PREY MODEL WITH CROWLEY–MARTIN FUNCTIONAL RESPONSE AND IMPULSES","authors":"Yaru Guo, Shulin Sun","doi":"10.1142/s0218339022500061","DOIUrl":"https://doi.org/10.1142/s0218339022500061","url":null,"abstract":"A stochastic non-autonomous one-prey two-predator model with Crowley–Martin functional response and impulses is proposed in this paper. First, by constructing the equivalent system without impulses, we investigate the existence and uniqueness of the global positive solution of the system. Second, by using Itô formula, strong law of large numbers and Chebyshev’s inequality, some sufficient conditions are established to ensure the extinction, non-persistence in the mean, persistence in the mean and stochastic permanence of the system. Third, we prove the system is globally attractive under some conditions. Finally, we choose different white noise intensities and impulsive parameters to illustrate the analytical results by numerical simulations.","PeriodicalId":54872,"journal":{"name":"Journal of Biological Systems","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46855563","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-11DOI: 10.1142/s0218339022500036
J. G. da Silva, I. C. D. da Silva, M. Adimy, Paulo Fernando de Arruda Mancera
Immunotherapy and targeted therapy are alternative treatments to differentiated thyroid cancer (DTC), which is usually treated with surgery and radioactive iodine. However, in advanced thyroid carcinomas, molecular alterations can cause a progressive loss of iodine sensitivity, thereby making cancer resistant to radioactive iodine-refractory (RAIR). In the treatment of cancer, tyrosine kinase inhibitors are administered to prevent the growth of cancer cells. One such inhibitor, lenvatinib, forms a targeted therapy for RAIR-DTC, while the immunotherapeutic pembrolizumab, a humanized antibody, prevents the binding of programmed cell death ligand 1 (PD-L1) to the PD-1 receptor. As one of the first studies on treatments for thyroid cancer with mathematical model involving immunotherapy and targeted therapy, we developed an ordinary differential system and tested variables such as concentration of lenvatinib and pembrolizumab, total cancer cells, and number of immune cells (i.e., T cells and natural killer cells). Analyzing local and global stability and the simulated action of drugs in patients with RAIR-DTC, revealed the combined effect of the targeted therapy with pembrolizumab. The scenarios obtained favor the combined therapy as the best treatment option, given its unrivaled ability to boost the immune system’s rate of eliminating tumor cells.
{"title":"THE EFFECT OF LENVATINIB AND PEMBROLIZUMAB ON THYROID CANCER REFRACTORY TO IODINE 131I SIMULATED BY MATHEMATICAL MODELING","authors":"J. G. da Silva, I. C. D. da Silva, M. Adimy, Paulo Fernando de Arruda Mancera","doi":"10.1142/s0218339022500036","DOIUrl":"https://doi.org/10.1142/s0218339022500036","url":null,"abstract":"Immunotherapy and targeted therapy are alternative treatments to differentiated thyroid cancer (DTC), which is usually treated with surgery and radioactive iodine. However, in advanced thyroid carcinomas, molecular alterations can cause a progressive loss of iodine sensitivity, thereby making cancer resistant to radioactive iodine-refractory (RAIR). In the treatment of cancer, tyrosine kinase inhibitors are administered to prevent the growth of cancer cells. One such inhibitor, lenvatinib, forms a targeted therapy for RAIR-DTC, while the immunotherapeutic pembrolizumab, a humanized antibody, prevents the binding of programmed cell death ligand 1 (PD-L1) to the PD-1 receptor. As one of the first studies on treatments for thyroid cancer with mathematical model involving immunotherapy and targeted therapy, we developed an ordinary differential system and tested variables such as concentration of lenvatinib and pembrolizumab, total cancer cells, and number of immune cells (i.e., T cells and natural killer cells). Analyzing local and global stability and the simulated action of drugs in patients with RAIR-DTC, revealed the combined effect of the targeted therapy with pembrolizumab. The scenarios obtained favor the combined therapy as the best treatment option, given its unrivaled ability to boost the immune system’s rate of eliminating tumor cells.","PeriodicalId":54872,"journal":{"name":"Journal of Biological Systems","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49650886","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-08DOI: 10.1142/s0218339022500024
P. Tiwari, Rajanish Kumar Rai, Rabindra Kumar Gupta, M. Martcheva, A. Misra
Media impact has significant effect on reducing the disease prevalence, meanwhile sanitation and awareness can control the epidemic by reducing the growth rate of bacteria and direct contacts with infected individuals. In this paper, we investigate the impacts of media and sanitation coverage on the dynamics of epidemic outbreak. We observe that the growth rate of social media advertisements carries out a destabilizing role, while the system regains stability if the baseline number of social media advertisements exceeds a certain threshold. The dissemination of awareness among susceptibles first destabilizes and then stabilizes the system. The disease can be wiped out if the baseline level of awareness or the rate of spreading global information about the disease and its preventive measures is too high. We obtain an explicit expression for the basic reproduction number [Formula: see text] and show that [Formula: see text] leads to the total eradication of infection from the region. To capture a more realistic scenario, we construct the forced delay model by seasonally varying the growth rate of social media advertisements and incorporating the time lag involved in reporting of total infective cases to the policy makers. Seasonal pattern in the growth rate of social media advertisements adds complexity to the system by inducing chaotic oscillations. For gradual increase in the delay in reported cases of infected individuals, the nonautonomous system switches finitely many times between periodic and chaotic states.
{"title":"Modeling the Control of Bacterial Disease by Social Media Advertisements: Effects of Awareness and Sanitation","authors":"P. Tiwari, Rajanish Kumar Rai, Rabindra Kumar Gupta, M. Martcheva, A. Misra","doi":"10.1142/s0218339022500024","DOIUrl":"https://doi.org/10.1142/s0218339022500024","url":null,"abstract":"Media impact has significant effect on reducing the disease prevalence, meanwhile sanitation and awareness can control the epidemic by reducing the growth rate of bacteria and direct contacts with infected individuals. In this paper, we investigate the impacts of media and sanitation coverage on the dynamics of epidemic outbreak. We observe that the growth rate of social media advertisements carries out a destabilizing role, while the system regains stability if the baseline number of social media advertisements exceeds a certain threshold. The dissemination of awareness among susceptibles first destabilizes and then stabilizes the system. The disease can be wiped out if the baseline level of awareness or the rate of spreading global information about the disease and its preventive measures is too high. We obtain an explicit expression for the basic reproduction number [Formula: see text] and show that [Formula: see text] leads to the total eradication of infection from the region. To capture a more realistic scenario, we construct the forced delay model by seasonally varying the growth rate of social media advertisements and incorporating the time lag involved in reporting of total infective cases to the policy makers. Seasonal pattern in the growth rate of social media advertisements adds complexity to the system by inducing chaotic oscillations. For gradual increase in the delay in reported cases of infected individuals, the nonautonomous system switches finitely many times between periodic and chaotic states.","PeriodicalId":54872,"journal":{"name":"Journal of Biological Systems","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49299304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-01DOI: 10.1142/s0218339022500073
S. Debnath, P. Majumdar, Sudeep Sarkar, U. Ghosh
In this paper, we have investigated global dynamics of a two-species food chain model with the Holling type III functional response that includes linear harvesting for the prey and nonlinear harvesting for the predator. The long-time continued existence of both species is discussed using uniform persistence theory. Stability of various equilibrium points is described in terms of model parameters. The local asymptotic stability of non-hyperbolic equilibrium points is determined with the help of center manifold theorem. Global behavior of solutions of the model system when both species are present is determined by considering the global properties of the coexistence equilibrium. Here, we have taken a comprehensive view by considering different bifurcations of co-dimension one and two and have discussed the importance of various model parameters on the system dynamics. The model system shows much more complex and realistic behavior compared to a model system without any harvesting, with constant harvesting or linear-yield harvesting of either or both of the species. Numerical simulations have been conducted to illustrate the theoretical findings.
{"title":"GLOBAL DYNAMICS OF A PREY–PREDATOR MODEL WITH HOLLING TYPE III FUNCTIONAL RESPONSE IN THE PRESENCE OF HARVESTING","authors":"S. Debnath, P. Majumdar, Sudeep Sarkar, U. Ghosh","doi":"10.1142/s0218339022500073","DOIUrl":"https://doi.org/10.1142/s0218339022500073","url":null,"abstract":"In this paper, we have investigated global dynamics of a two-species food chain model with the Holling type III functional response that includes linear harvesting for the prey and nonlinear harvesting for the predator. The long-time continued existence of both species is discussed using uniform persistence theory. Stability of various equilibrium points is described in terms of model parameters. The local asymptotic stability of non-hyperbolic equilibrium points is determined with the help of center manifold theorem. Global behavior of solutions of the model system when both species are present is determined by considering the global properties of the coexistence equilibrium. Here, we have taken a comprehensive view by considering different bifurcations of co-dimension one and two and have discussed the importance of various model parameters on the system dynamics. The model system shows much more complex and realistic behavior compared to a model system without any harvesting, with constant harvesting or linear-yield harvesting of either or both of the species. Numerical simulations have been conducted to illustrate the theoretical findings.","PeriodicalId":54872,"journal":{"name":"Journal of Biological Systems","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46422973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-01DOI: 10.1142/s021833902250005x
R. Gupta, Dinesh K. Yadav
The role of scavengers, which consume the carcasses of predators along with predation of the prey, has been ignored in comparisons to herbivores and predators. It has now become a topic of high interest among researchers working with food-web systems of prey–predator interactions. The food-web considered in these works contains prey, predators, and scavengers as the third species. In this work, we attempt to study a food-web model of these species in the presence of the multiplicative Allee effect and harvesting. It is observed that this makes the model more complex in the form of multiple co-existing steady states. The conditions for the existence and local stability of all possible steady states of the proposed system are analyzed. The global stability of the steady state lying on the x-axis and the interior steady state have been discussed by choosing suitable Lyapunov functions. The existence conditions for saddle-node and Hopf bifurcations are derived analytically. The stability of Hopf bifurcating periodic solutions with respect to both Allee and harvesting constants is examined. It is also observed that multiple Hopf bifurcation thresholds occur for harvesting parameters in the case of two co-existing steady states, which indicates that the system may regain its stability. The proposed model is also studied beyond Hopf bifurcation thresholds, where we have observed that the model is capable of exhibiting period-doubling routes to chaos, which can be controlled by a suitable choice of Allee and harvesting parameters. The largest Lyapunov exponents and sensitivity to initial conditions are examined to ensure the chaotic nature of the system.
{"title":"ROLE OF ALLEE EFFECT AND HARVESTING OF A FOOD-WEB SYSTEM IN THE PRESENCE OF SCAVENGERS","authors":"R. Gupta, Dinesh K. Yadav","doi":"10.1142/s021833902250005x","DOIUrl":"https://doi.org/10.1142/s021833902250005x","url":null,"abstract":"The role of scavengers, which consume the carcasses of predators along with predation of the prey, has been ignored in comparisons to herbivores and predators. It has now become a topic of high interest among researchers working with food-web systems of prey–predator interactions. The food-web considered in these works contains prey, predators, and scavengers as the third species. In this work, we attempt to study a food-web model of these species in the presence of the multiplicative Allee effect and harvesting. It is observed that this makes the model more complex in the form of multiple co-existing steady states. The conditions for the existence and local stability of all possible steady states of the proposed system are analyzed. The global stability of the steady state lying on the x-axis and the interior steady state have been discussed by choosing suitable Lyapunov functions. The existence conditions for saddle-node and Hopf bifurcations are derived analytically. The stability of Hopf bifurcating periodic solutions with respect to both Allee and harvesting constants is examined. It is also observed that multiple Hopf bifurcation thresholds occur for harvesting parameters in the case of two co-existing steady states, which indicates that the system may regain its stability. The proposed model is also studied beyond Hopf bifurcation thresholds, where we have observed that the model is capable of exhibiting period-doubling routes to chaos, which can be controlled by a suitable choice of Allee and harvesting parameters. The largest Lyapunov exponents and sensitivity to initial conditions are examined to ensure the chaotic nature of the system.","PeriodicalId":54872,"journal":{"name":"Journal of Biological Systems","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44135260","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：