TY - JOUR
T1 - A predator–prey model with Crowley–Martin functional response
T2 - A nonautonomous study
AU - Tripathi, Jai Prakash
AU - Bugalia, Sarita
AU - Tiwari, Vandana
AU - Kang, Yun
N1 - Funding Information:
The research work of first author (Jai Prakash Tripathi) is supported by Science and Engineering Research Board (SERB), India (File no. ECR/2017/002786) and UGC‐BSR Research Start‐Up‐Grant, India (No. F.30‐356/2017(BSR)). The research work of Sarita Bugalia is supported by the Council of Scientific & Industrial Research (CSIR), India (File no. 09/1131(0025)/2018‐EMR‐I). The work of Y.K. is also partially supported by NSF‐DMS (1716802); NSF‐IOS/DMS (1558127), and The James S. McDonnell Foundation 21st Century Science Initiative in Studying Complex Systems Scholar Award (UHC Scholar Award 220020472).
Funding Information:
The research work of first author (Jai Prakash Tripathi) is supported by Science and Engineering Research Board (SERB), India (File no. ECR/2017/002786) and UGC-BSR Research Start-Up-Grant, India (No. F.30-356/2017(BSR)). The research work of Sarita Bugalia is supported by the Council of Scientific & Industrial Research (CSIR), India (File no. 09/1131(0025)/2018-EMR-I). The work of Y.K. is also partially supported by NSF-DMS (1716802); NSF-IOS/DMS (1558127), and The James S. McDonnell Foundation 21st Century Science Initiative in Studying Complex Systems Scholar Award (UHC Scholar Award 220020472).
Publisher Copyright:
© 2020 The Authors. Natural Resource Modeling published by Wiley Periodicals LLC
PY - 2020/11/1
Y1 - 2020/11/1
N2 - We investigate a nonautonomous predator–prey model system with a Crowley–Martin functional response. We perform rigorous mathematical analysis and obtain conditions for (a) global attractivity and permanence in the form of integrals which improve the traditional conditions obtained by using bounds of involved parameters; and (b) the existence of periodic solutions applying continuation theorem from coincidence degree theory which has stronger results than using Brouwer fixed point theorem. Our result also indicates that the global attractivity of periodic solution is positively affected by the predator's density dependent death rate. We employ partial rank correlation coefficient method to focus on how the output of the model system analysis is influenced by variations in a particular parameter disregarding the uncertainty over the remaining parameters. We discuss the relations between results (permanence and global attractivity) for autonomous and nonautonomous systems to get insights on the effects of time-dependent parameters. Recommendations for Resource Managers: The natural environment fluctuates because of several factors, for example, mating habits, food supplies, seasonal effects of weathers, harvesting, death rates, birth rates, and other important population rates. The temporal fluctuations in physical environment (periodicity) plays a major role in community and population dynamics along with the impacts of population densities. Periodic system may suppress the permanence of its corresponding autonomous system with parameters being the averages of periodic parameters. As the human needs crosses a threshold level, then we require to observe the sustainability of resources of the associated exploited system. Therefore, the concept of stability and permanence become our main concern in an exploited model system (system with harvesting). The mutual interference at high prey density may leave negative effect on the permanence of the system. In harvested system, permanence becomes an important issue because if we harvest too many individuals then species may be driven to extinction. Interestingly, in many biological/agricultural systems, harvesting (due to fishing in marine system, hunting or disease) of a particular species/crop can only be more beneficial at certain times (e.g., the time and stage of harvest of a particular crop play greater role in its production and hence the particular crop is many times harvested at its physiological maturity or at harvest maturity).
AB - We investigate a nonautonomous predator–prey model system with a Crowley–Martin functional response. We perform rigorous mathematical analysis and obtain conditions for (a) global attractivity and permanence in the form of integrals which improve the traditional conditions obtained by using bounds of involved parameters; and (b) the existence of periodic solutions applying continuation theorem from coincidence degree theory which has stronger results than using Brouwer fixed point theorem. Our result also indicates that the global attractivity of periodic solution is positively affected by the predator's density dependent death rate. We employ partial rank correlation coefficient method to focus on how the output of the model system analysis is influenced by variations in a particular parameter disregarding the uncertainty over the remaining parameters. We discuss the relations between results (permanence and global attractivity) for autonomous and nonautonomous systems to get insights on the effects of time-dependent parameters. Recommendations for Resource Managers: The natural environment fluctuates because of several factors, for example, mating habits, food supplies, seasonal effects of weathers, harvesting, death rates, birth rates, and other important population rates. The temporal fluctuations in physical environment (periodicity) plays a major role in community and population dynamics along with the impacts of population densities. Periodic system may suppress the permanence of its corresponding autonomous system with parameters being the averages of periodic parameters. As the human needs crosses a threshold level, then we require to observe the sustainability of resources of the associated exploited system. Therefore, the concept of stability and permanence become our main concern in an exploited model system (system with harvesting). The mutual interference at high prey density may leave negative effect on the permanence of the system. In harvested system, permanence becomes an important issue because if we harvest too many individuals then species may be driven to extinction. Interestingly, in many biological/agricultural systems, harvesting (due to fishing in marine system, hunting or disease) of a particular species/crop can only be more beneficial at certain times (e.g., the time and stage of harvest of a particular crop play greater role in its production and hence the particular crop is many times harvested at its physiological maturity or at harvest maturity).
KW - almost periodic solution
KW - coincidence degree
KW - functional response
KW - global attractivity
KW - periodic solution
KW - sensitivity analysis
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U2 - 10.1111/nrm.12287
DO - 10.1111/nrm.12287
M3 - Article
AN - SCOPUS:85093520193
SN - 0890-8575
VL - 33
JO - Natural Resource Modeling
JF - Natural Resource Modeling
IS - 4
M1 - e12287
ER -