Bounded perturbation for evolution equations with a parameter &amp; application to population dynamics

Emile Franc Doungmo Goufo

doi:10.3934/dcdss.2020177

Article Contents

2021, Volume 14, Issue 7: 2137-2150. Doi: 10.3934/dcdss.2020177

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Bounded perturbation for evolution equations with a parameter & application to population dynamics

Emile Franc Doungmo Goufo^,

Department of Mathematical Sciences, University of South Africa, Florida, 0003, South Africa

^* Corresponding author: franckemile2006@yahoo.ca
^* Corresponding author: franckemile2006@yahoo.ca

Received: April 2019

Revised: September 2020

Early access: May 2021

Published: July 2021

This work was partially supported by the grant No: 105932 from the National Research Foundation (NRF) of South Africa

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Abstract

Evolution equations using derivatives of fractional order like Caputo's derivative or Riemann-Liouville's derivative have been intensively analyzed in numerous works. But the classical bounded perturbation theorem has been proven not to be in general true for these models, especially for solution operators of evolution equations with fractional order derivative $ \alpha $ less than $ 1 $ ($ 0<\alpha<1 $), as shown by the example in the next section. This paper proposes an alternative way of dealing with this issue. We make use of the conventional time derivative with a new parameter to show the perturbations by bounded linear operators for linear evolution equations when the derivative order is less than one. The new parameter which happens to be fractional, characterizes the so-called $ \beta $-derivative. Its fractional order parameter allows the use of concepts like revamped time to provide a relation between both strongly continuous two-parameter solution operators involved in the perturbation process. To validate the theory, we use an application to population dynamics and perform some numerical simulations that reveal some consistency with the expected results.

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Mathematics Subject Classification: Primary: 34D10, 92D25; Secondary: 34A08, 46F30.

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Figure 1. Numerical solutions showing the time evolution of the history intervals $ [-\tau, 0], \ \ \ \tau>0, $

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Figure 2. Numerical solutions showing evolution dynamics in the time interval [0, 25] for solutions to (33)-(34). Different delays are considered and similar trajectories are shown in both figures (integer and pure fractional order)

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