Stability Analysis for a Delayed Lotka-Volterra System with Harvesting and Stocking Rates in Parameter Space

In this paper, we focus on the stability of a delayed Lotka-Volterra type predator-prey system with harvesting rate K
1
and stocking rate K
2
. All the system parameters (K
1
, K
2
) and the delay τ are treated as free parameters. Our objective is to study the stability in the entire (K
1
, K
2
, τ)-space. First, we consider the case when (K
1
, K
2
) is fixed and analyze the stability along the whole τ axis. Such a problem, which is called the complete stability problem w.r.t. delay, can be solved by addressing the corresponding auxiliary characteristic roots. The stability analysis is then extended to the scenario with additional free system parameters (K
1
, K
2
). We can study the complete real root classification for the auxiliary characteristic equation. As a result, we are able to characterize the stability property over the entire (K
1
, K
2
, τ)-space. The approach presented in this paper provides a new way to design (K
1
, K
2
) such that the system is in a stable state. Finally, some representative examples are given to demonstrate the effectiveness of the approach.