Plankton-oxygen dynamics in the context of climate change: a fractional model with a probability density function approach

Abstract : Climate change due to global warming is a grave issue that threatens not only humans, but 
also numerous ecosystems that are vital for their survival. Of these, the marine ecosystem which is 
responsible for producing the greatest amount of atmospheric oxygen. To better understand the effects of 
climate change on this ecosystem, we study the plankton-oxygen dynamics with a fractional nonlinear 
model on the form d𝛼𝑥(𝑡)/d𝑡𝛼 = 𝐴𝑥(𝑡) + 𝑓(𝑥(𝑡)), where 𝐴 is a diagonal matrix of order 3 and 𝑓(𝑥(𝑡)) 
is a nonlinear function. First, we rigorously formulate the model. Then, under suitable conditions, we 
determine a Lipschitz constant of 𝑓. Existence and uniqueness of the solution, in addition to the 
wellposedness of the problem, are studied using a fractional formula with a probability density function, 
with the aid of a generalized Grönwall’s inequality. Furthermore, wellposedness of the problem is 
investigated using the continuous dependence on the initial condition.