A note on Arzela–Ascoli's lemma in almost periodic problems

Zhou, Qing; Shao, Jianying

doi:10.1002/mma.3988

Cited by 5 publications

(1 citation statement)

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“…We have also discussed more general case than the existence of periodic solution that is, we established the existence of a positive almost periodic solution (refer the Theorem 13). It is important to mention that this particular proof of existence of unique almost periodic solution do not make use of Arzela–Ascoli's Lemma (Rudin, 2008; Zhou & Shao, 2017). …”

Section: Discussionmentioning

confidence: 99%

A predator–prey model with Crowley–Martin functional response: A nonautonomous study

Tripathi

Bugalia

Tiwari³

et al. 2020

Natural Resource Modeling

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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).

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Section: Discussionmentioning

confidence: 99%