No abstract

On the set of positive real numbers, multiplication, represented by ⊕, is considered as an operation associated with the notion of sum, and the operation a ⨀ b = aln(b) represents the meaning of the traditional multiplication. The triple (R+, ⊕,⨀) forms an ordered and complete field in which derivative and integration operators are defined analogously to the Differential and Integral Calculus. In this article, we present the proportional arithmetic and we construct the theory of ordinary proportional differential equations. A proportional version of Gronwall inequality, Gompertz’s function, the q-Periodic functions, proportional heat, and wave equations as well as a proportional version of Fourier’s series are presented. Furthermore, a non-Newtonian logistic growth model is proposed.

Environmental temperature and body size influence the life cycle of the species, with consequences for population size. In addition, it has been reported that increased temperature can lead to a decrease in body size. In the context of a resource‐stock, whose abundance is diminished by the action of an endothermic predator and also by small‐scale fishing activity, we analysed a Schaefer‐type fishery model that incorporates parametric variables associated with thermal performance, metabolic theory, and warming. We project the biomass of the resource in a thermal tolerance range with an increasing temperature trend obtained from current data. In the short term there could be an increase in biomass. However, over time the stock will decline rapidly, in association with the intensity of temperature increase and fishing effort. Recommendations for Resource Managers Temperature increments towards tolerance limits will generate that fish resources will eventually collapse locally. In the meantime, latitudinal variation in the physiological sensitivity of organisms will affect the outcome of the fishing effort differentially. Planning for more sustainable harvesting requires an ecophysiological understanding of both the resource and the species with which it interacts. Simple models contribute to the formulation of strategies aimed to guide responsible monitoring systems, where theoretical, observational, and experimental information is highly relevant.

Predation is a biological interaction that influences demographic patterns by modifying community structure. In the current ecological crisis, there is a need to better understand the conditions of coexistence between predators, prey and their resources. The body size is considered a key feature to explain community-scale phenomena, energetic, and evolutionary constraints. This raises the question of how species body size directly or indirectly affects the demographic patterns that enable coexistence. Considering the above, we conducted a theoretical study that implements a Rosenzweig-MacArthur type model, which represents a three-level chain that integrates body sizes and includes a Holling type I functional response. In this model, we characterize coexistence through body size-dependent net reproductive rates. Our results suggest that the body sizes of consumer species strongly affect the size-density relations and energy requirements. We obtain the negative relationship between body size and density of intermediate consumers and discuss the energy equivalence rule. Furthermore, larger predators have a more significant impact on the intensity of the trophic cascade than smaller predators. Finally, we discuss potential extensions and applications of our modeling approach.