Modelling the dynamics of two political parties in the presence of switching
Abstract
This paper generalizes the model proposed by Misra, by considering switching between political parties. In the model proposed, the movements of members from political party B to political party C and vice versa, are considered but the net movement is considered by assuming that \(\theta _1\theta _2=\theta\) (a constant), which implies that the movement of members is either from party B to party C or from party C to party B. In this paper we remodel these movements through switching functions to capture how individuals switch between parties. The results provide a more comprehensive synopsis of the dynamics between two political parties.
Keywords
Political parties Modelling Switching function Steady states SimulationsBackground
In ecology, the term switching was first coined by Murdoch in 1969, to describe a scenario where a predator predominantly eats the most common type of prey, see Murdoch (1969) and is often accompanied by a change in the habitat Khan (2000). Prey switching however happens when a predator’s preference for a particular type of prey increases as the prey increases in abundance. Any display by a predator, of prey switching behaviour, can significantly affect the stability of the system, coexistence of prey species and evolutionary diversification. Switching can however promote coexistence between prey species Abrams and Matsuda (2003). A classical example is the case where prey switching causes low predation for rare prey, thus aiding prey refugia that often leads to coexistence Gentleman et al. (1990).
More often than not, political parties compete for membership. Members often switch between political parties as preferences change, often as a result of change of leadership, policies and perceived gains Fieldhouse et al. (2007), Petersen (1991), Schofield and Sened (2005), Romero et al. (2009). This paper is motivated by the work in Misra (2012). A closer loot at the work in Misra (2012) shows that there were simplifying assumptions that made the mathematical model tractable but overlooking some essential elements such as switching. The parameters \(\theta _1\) and \(\theta _2\) model movements between political parties B and C. The net shifting of members \(\theta =\theta _1\theta _2\) is considered to be constant resulting in a unidirectional movement of members from B to C and vice versa. In this paper, we relook at this assumption by introducing switching functions whose parameters are endogenous to the system.
The paper is arranged as follows: in “The Misra model” section, we generalize the Misra model by including switching functions. The stabilities of the steady states are presented in “Stability of steady states” section and the paper is concluded in “Conclusion” section.
The Misra model
Here \(\theta (t,b,c)\) can either be positive or negative, thus allowing individuals to switch between political parties. Just as in Misra (2012), system (3) has four equilibria, a party free equilibrium \(E_0 = (0,0),\) single party equilibria \(E_1 = (1\dfrac{\mu }{\beta _1},0)\) and \(E_2 = (0,1\dfrac{\mu }{\beta _2}),\) whose existence is subject to \(\beta _1 > \mu\) and \(\beta _2 > \mu\) respectively and the interior equilibrium. Unlike in Misra (2012), the interior equilibrium is only unique for the case \(\theta _1(b) =\theta _2(c)\).
 (1)
The case \(\theta (t,b,c)<0.\) This is the case where members leave political party C for B, for all \(t<ts\).
 (2)
The case \(\theta (t,b,c) = 0.\)
 (3)
The case \(\theta (t,b,c)>0.\) This is the case considered in Misra (2012). This is the case where members leave political party B for C, for all \(t>ts\) where ts is the time at which the switch occurs.
Stability of steady states

If \(\beta _1 \ne \beta _2\), then the interior equilibrium \(E(b^*,c^*)\) does not exist.
 If \(\beta _1 = \beta _2\), then the interior equilibrium is stable and is a straight line satisfying the equation$$\begin{aligned} \left\{ b^*, c^* \in [0,1]b^* + c^* = 1  \dfrac{\mu }{\beta _1}\right\} . \end{aligned}$$
A closer look at the Fig. 6 shows that there exists a time interval \(t<ts\) where members of political party C leave for political party B when \(\theta (t)<0\). After that members of political party B leave for party C before both become stable over the time. The corresponding interior equilibrium is shown in Fig. 7.
Conclusion
In this paper, we remodelled switching between political parties in the model formulated in Misra (2012). This was achieved by removing the constraint that the difference between the net rates of movement between the two political parties be sign definite. We defined two switching functions that depend on the size of each political party and some parameters. These functions generalize the Misra paper in which the net movement was assumed to be unidirectional, or in favour of a given political party. In addition to some results obtained in Misra (2012), additional information regarding how the behaviour of the population size is dependent on the switching parameters is demonstrated.
The inclusion of switching functions in this paper improved the Misra (2012) model. There are further aspects that can be considered in future. Among these we mention the possibility of including individual preferences in choosing a political party. Another aspect will be the improvement of the model by considering a non constant population. An interesting aspect to consider is the age structured model, in view of the fact that political parties often target the youths for the future sustainability of the parties. There is however a trade off between mathematical tractability and realism. Finally, one can also look at how media companies influence the dynamics of political parties.
Notes
Authors' contributions
FN was instrumental in the conception and design of the model. TYA carried out the mathematical analysis and numerical simulations. GM did the mathematical analysis and all authors participated in writing and interpretation of numerical results of the manuscript. All authors read and approved the final manuscript.
Acknowledgements
The first author acknowledges the support of Stellenbosch University in the production of the manuscript. The second author acknowledges the support of AIMS, Ghana. The third author acknowledges the support of the University of Zimbabwe.
Competing interests
The authors declare that they have no competing interests.
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