Steady-state solutions and stability of a mathematical model of two interacting legumes

Abstract

In this paper, we studied a deterministic population model with a mass action law simplification. The main aim is to determine the steady-state solutions and their stability. From our previous study, we have used the selection method of penalty functions to select the estimated best-fit values of the model parameters a, b, d, f, while the precise values of the parameters c and e are assumed. The first model equations using the 1-norm selection method have four steady-state solutions (0, 0), (0, 3.3534), (3.2599, 0) and (3.1908, 0.9543). The second model equations using the 2-norm selection method have four steady-state solutions (0, 0), (0, 3.6860), (3.0000, 0) and (1.1341, 3.4985) while the third model equations using the infinity-norm selection method have four steady-state solutions (0, 0), (0, 4.0545), (2.7898, 0) and (0.5871, 3.9478). Irrespective of the type of parameter selection method, we observe that the first three steady-state solutions are unstable while the unique positive co-existence steady-state solutions are said to be stable. Computer simulations were used to illustrate our theory. However, the question of stabilizing the unstable steady-state solutions which we have found in this study remains to be numerically answered. This level of analysis will be attempted in our next simulation study.

Key words: Steady-State Solutions, Stability, Mass Action Law.