Population growth and competition models with decay and competition consistent delay

• Published in: Journal of mathematical biology Vol. 84; no. 5; p. 39
• Main Authors: Lin, Chiu-Ju, Hsu, Ting-Hao, Wolkowicz, Gail S. K.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2022; Springer Nature B.V
• Subjects: Applications of Mathematics; Competition; Delay; Differential equations; Dynamic stability; Ecosystem; Growth rate; Mathematical and Computational Biology; Mathematics; Mathematics and Statistics; Models, Biological; Offspring; Oscillations; Population; Population decline; Population Density; Population Dynamics; Population Growth; Species; Species extinction; Discrete delay differential equations; Local and global dynamics; Logistic growth; Chain transitive sets; 92D25 Population dynamics; Lotka–Volterra competition; 92D40 Ecology; Extinction threshold; 37N25 Dynamical systems in biology
• ISSN: 0303-6812; 1432-1416
• DOI: 10.1007/s00285-022-01741-3

We derive an alternative expression for a delayed logistic equation in which the rate of change in the population involves a growth rate that depends on the population density during an earlier time period. In our formulation, the delay in the growth term is consistent with the rate of instantaneous decline in the population given by the model. Our formulation is a modification of Arino et al. (J Theor Biol 241(1):109–119, 2006) by taking the intraspecific competition between the adults and juveniles into account. We provide a complete global analysis showing that no sustained oscillations are possible. A threshold giving the interface between extinction and survival is determined in terms of the parameters in the model. The theory of chain transitive sets and the comparison theorem for cooperative delay differential equations are used to determine the global dynamics of the model. We extend our delayed logistic equation to a system modeling the competition between two species. For the competition model, we provide results on local stability, bifurcation diagrams, and adaptive dynamics. Assuming that the species with shorter delay produces fewer offspring at a time than the species with longer delay, we show that there is a critical value, τ ∗ , such that the evolutionary trend is for the delay to approach τ ∗ .