A Financial Prey-predator Model with Infection in the Predator

A modified predator-prey model is proposed with logistic growth in both prey and predator populations and an infection in the predator population. This model uses ideas from the original biological Lotka-Volterra model in an attempt to imitate financial predation. Potential investors, the prey, interact with financial experts, the predators, who provide advice on purchasing financial instruments and investments. Some of these experts are honest while others are ‘infected’, in that, a portion of them attempt to deceive clients into making irrational investments for their own benefit, incurring losses to the client. Stability and Hopf bifurcation analyses are discussed analytically. The results have been verified using numerical simulations in MATLAB. Using different datasets, the study shows that variation of different parameters can affect the stability of the system and the co-existence of potential investors and financial experts over time.

Aims: To determine regions of stability for the model by varying parameter values in simulated datasets.

Study Design: Stability Analysis – Analytical and Numerical.

Methodology: The model is first studied analytically by solving for the positive, interior equilibrium point (once it exists) when the differential equations are solved simultaneously. Stability conditions are defined based on these results using the Routh – Hurwitz criteria. These results are verified numerically in MATLAB as an initial value problem, integrating with slight perturbation around the equilibrium point. Each parameter in the dataset is varied one at a time. The regions where the model remains stable for a particular parameter are recorded. The Hopf bifurcation points, where there are stability changes, are also noted and graphical simulations are produced using time series and three-dimensional (3D) plots for particular stable and unstable scenarios. The effect of three different pairs of financial expert - investor persuasion rates on the populations of investors, is also shown.

Results: Hopf bifurcation parameters for three simulated datasets were found to be associated with the investment persuasion rate, investor-expert interaction rates, expert interaction rate, expert interference constant and growth rate of experts. In particular, the population of potential investors increased drastically when the investor persuasion rate for dishonest investors was much greater than that for honest investors for the given dataset.

Conclusion: Variation of parameter values in the model allow for the analysis of model stability conditions. These enhance the creation of detection mechanisms to control the ‘infection’ of investment fraud among financial experts.