Skip to main content

Two-species competing population dynamics with the population-dependent environmental capacities under random disturbance


We propose and analyze a stochastic competing two-species population dynamics model subject to jump and continuous correlated noises. Competing benthic algae population dynamics in river environment, which is an important engineering problem, motivates this new model. The model is a system of stochastic differential equations having a characteristic that the two populations are competing with each other through the environmental capacities; an increase in one population decreases the other’s environmental capacity. Unique existence of the uniformly bounded strong solution is proven, and attractors of the solutions are identified depending on the parameter values. The Kolmogorov’s backward equation associated with the population dynamics is formulated and its unique solvability in a Banach space with a weighted norm is discussed. A novel uncertain correlation case is also analyzed in the framework of viscosity solutions. Numerical computation results using a finite difference scheme and a Monte-Carlo method are presented to deeper analyze the model. Our analysis results can be utilized for establishment of a foundation for modeling, analysis and control of the competing population dynamics.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9


Download references


Kurita Water and Environment Foundation Grant No. 19B018, and a grant for ecological survey of a life history of the landlocked ayu Plecoglossus altivelis altivelis from the Ministry of Land, Infrastructure, Transport and Tourism of Japan support this research. This research was carried out under a support of the research fund for young researchers in Shimane University.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Hidekazu Yoshioka.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yoshioka, H. Two-species competing population dynamics with the population-dependent environmental capacities under random disturbance. Theory Biosci. 139, 279–297 (2020).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • River environment
  • Algae bloom
  • Stochastic differential equation
  • Population-dependent environmental capacity
  • Kolmogorov’s backward equation
  • Weak solutions