Effects of Age-Dependent Competition During Immature Developmental Stages on Species Population

¹ Department of Mathematics, National Taiwan Normal University, No. 88, Sec. 4, Ting-Chou Rd., Taipei 116, Taiwan
² Department of Mathematics, National Kaohsiung Normal Universit, No. 62, Shenzhong Rd., Kaohsiung City 82444, Taiwan
³ Department of Mathematical Sciences, National Chengchi University, No. 64, Sec. 2, Zhinan Rd., Wenshan District, Taipei City 11605, Taiwan

^* Corresponding author: cycheng@mail.nknu.edu.tw

Received: 14 February 2025
Accepted: 4 July 2025

Abstract

This study extends the model proposed by Lin et al. [J. Math. Biol. 84 (2022) 39] by incorporating an age-dependent competition within species into the logistic equation. The proposed model is a differential equation with an integral delay, reflecting variations in competition. In the single-species model, sustained oscillations occur due to Hopf bifurcations related to the immature period or competition variation index. A threshold dynamics can be determined, either global convergence to a trivial equilibrium or the uniform persistence of positive solutions. Additionally, we can demonstrate the dynamics of convergence towards a positive equilibrium under another criterion by using the convergence property of a class of linear functional differential equations. In the two-species model, we certify that the population dynamics converge toward the dominant single-species equilibria when one species can survive in the environment without competitors while the other cannot. If a positive equilibrium exists with strong competition between the two species, we numerically observe bistable dynamics. On the other hand, when competition between the two species is low, we numerically observe multiple stability switches in the coexistence state. Additionally, we examine how the maturation time affects the dynamics of invader populations under steady or oscillatory conditions of the resident species.