{"title":"CMMSE: New Properties of Auto-Wave Solutions in Activator-Inhibitor Reaction-Diffusion Systems With Fractional Derivatives","authors":"Bohdan Datsko, Vasyl Gafiychuk","doi":"10.1002/mma.10672","DOIUrl":"https://doi.org/10.1002/mma.10672","url":null,"abstract":"<div>\u0000 \u0000 <p>In this article, we analyze new properties of auto-wave solutions in fractional reaction-diffusion systems. These new properties arise due to a change in fractional derivative order and do not occur in systems with classical derivatives. It is shown that the stability of steady-state solutions and their evolution are mainly determined by the eigenvalue spectrum of a linearized system and the fractional derivative order. It is also demonstrated that the basic properties of auto-wave solutions in fractional-order systems can essentially differ from those in standard systems. The results of the linear stability analysis are confirmed by computer simulations of the generalized fractional van der Pol–FitzHugh–Nagumo mathematical model. A common picture of possible instabilities and auto-wave solutions in time-fractional two-component activator-inhibitor systems is presented.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 6","pages":"6292-6302"},"PeriodicalIF":2.1,"publicationDate":"2024-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143622342","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Local Well-Posedness of Kolmogorov'S Two-Equation Model of Turbulence in Fractional Sobolev Spaces","authors":"Przemysław Kosewski","doi":"10.1002/mma.10643","DOIUrl":"https://doi.org/10.1002/mma.10643","url":null,"abstract":"<div>\u0000 \u0000 <p>We study Kolmogorov's two-equation model of turbulence on \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 <annotation>$$ d $$</annotation>\u0000 </semantics></math>-dimensional torus. First, the local existence of the solution with the initial data from non-homogeneous fractional Sobolev spaces (Bessel potential spaces) \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>s</mi>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$$ {H}^s $$</annotation>\u0000 </semantics></math> with \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>s</mi>\u0000 <mo>></mo>\u0000 <mfrac>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 </mfrac>\u0000 </mrow>\u0000 <annotation>$$ s>frac{d}{2} $$</annotation>\u0000 </semantics></math> is proven using energy methods. Next, we show that solutions are unique in the class of solutions guaranteed by the local existence theorem.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 5","pages":"5872-5895"},"PeriodicalIF":2.1,"publicationDate":"2024-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143595695","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Seasonal Stochasticity Drives Phytoplankton–Zooplankton Dynamics","authors":"Nazmul Sk, Subarna Roy, Pankaj Kumar Tiwari","doi":"10.1002/mma.10651","DOIUrl":"https://doi.org/10.1002/mma.10651","url":null,"abstract":"<div>\u0000 \u0000 <p>In this study, we examine a deterministic model incorporating refuge, additional food sources, and toxins. The existence and stability of positive equilibria and the bifurcation analyses are discussed. Our numerical outcomes entail that increased zooplankton growth due to additional food leads to the disappearance of phytoplankton species, while a significant drop in nutrient levels results in zooplankton extinction within the ecosystem. Notably, the refuge by phytoplankton has a tendency to terminate the persistent oscillations and stabilize the system. As seasonal stochasticity significantly influences the dynamics of planktonic system, so we introduce seasonality into environmental noise and certain model parameters. In this case, we analyze both the regularity and the dichotomy between persistence and extinction. The numerical evidences demonstrate periodic solutions, strong/weak persistence, and plankton extinctions resulting from stochasticity and/or seasonality. Furthermore, the seasonally forced noise, intriguingly, has the capacity to exert control over hyperchaos, yielding a distinctive pattern in plankton populations.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 5","pages":"5998-6018"},"PeriodicalIF":2.1,"publicationDate":"2024-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143595693","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Approximation Properties of \u0000(λ,μ)-Bernstein-Durrmeyer Operators","authors":"Qing-Bo Cai, Guorong Zhou","doi":"10.1002/mma.10647","DOIUrl":"https://doi.org/10.1002/mma.10647","url":null,"abstract":"<div>\u0000 \u0000 <p>In this manuscript, a new kind of (\u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 <mo>,</mo>\u0000 <mspace></mspace>\u0000 <mi>μ</mi>\u0000 </mrow>\u0000 <annotation>$$ lambda, kern0.3em mu $$</annotation>\u0000 </semantics></math>)-Bernstein-Durrmeyer operators is introduced. A Korovkin-type approximation theorem is obtained, the rate of convergence is investigated by using the modulus of smoothness, Lipschitz continuous function, and Steklov mean, a Voronovskaja asymptotic formula is established, and graphical representations and numerical examples are also presented to compare the newly defined ones with other forms.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 5","pages":"5946-5953"},"PeriodicalIF":2.1,"publicationDate":"2024-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143595643","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}