ABSTRACT
The interaction among phytoplankton and zooplankton is one of the most important processes in ecology. Discrete-time mathematical models are commonly used for describing the dynamical properties of phytoplankton and zooplankton interaction with nonoverlapping generations. In such type of generations a new age group swaps the older group after regular intervals of time. Keeping in observation the dynamical reliability for continuous-time mathematical models, we convert a continuous-time phytoplankton-zooplankton model into its discrete-time counterpart by applying a dynamically consistent nonstandard difference scheme. Moreover, we discuss boundedness conditions for every solution and prove the existence of a unique positive equilibrium point. We discuss the local stability of obtained system about all its equilibrium points and show the existence of Neimark-Sacker bifurcation about unique positive equilibrium under some mathematical conditions. To control the Neimark-Sacker bifurcation, we apply a generalized hybrid control technique. For explanation of our theoretical results and to compare the dynamics of obtained discrete-time model with its continuous counterpart, we provide some motivating numerical examples. Moreover, from numerical study we can see that the obtained system and its continuous-time counterpart are stable for the same values of parameters, and they are unstable for the same parametric values. Hence the dynamical consistency of our obtained system can be seen from numerical study. Finally, we compare the modified hybrid method with old hybrid method at the end of the paper.
ABSTRACT
Based on a recent paper of Beg and Pathak (Vietnam J. Math. 46(3):693-706, 2018), we introduce the concept of H q + -type Suzuki multivalued contraction mappings. We establish a fixed point theorem for this type of mappings in the setting of complete weak partial metric spaces. We also present an illustrated example. Moreover, we provide applications to a homotopy result and to an integral inclusion of Fredholm type. Finally, we suggest open problems for the class of 0-complete weak partial metric spaces, which is more general than complete weak partial metric spaces.
ABSTRACT
A Lyapunov-type inequality is established for the anti-periodic fractional boundary value problem ( C D a α , ψ u ) ( x ) + f ( x , u ( x ) ) = 0 , a < x < b , u ( a ) + u ( b ) = 0 , u ' ( a ) + u ' ( b ) = 0 , where ( a , b ) ∈ R 2 , a < b , 1 < α < 2 , ψ ∈ C 2 ( [ a , b ] ) , ψ ' ( x ) > 0 , x ∈ [ a , b ] , D a α , ψ C is the ψ-Caputo fractional derivative of order α, and f : [ a , b ] × R â R is a given function. Next, we give an application of the obtained inequality to the corresponding eigenvalue problem.
