M. S. Santhanam and Aanjaneya Kumar
10.3 Lotka-Volterra (LV) Model: Predator-Prey
For a very long time, researchers have investigated the famous prey-predator model, i.e., Lotka-Volterra system of equations for analyzing endogenous oscillatory behav- ior of coupled nonlinear equations. Sirohi et al. (2015) recently introduced environ- mental noise in such a model and studied the corresponding effects on spatio-temporal pattern formation. Due to the addition of environmental noise, random fluctuations have been observed for the fundamental variables characterizing the predator-prey system, viz. carrying capacity, intensity of intra- and inter-species competition rates, birth and death rates, and predation rates. In particular, they study a simple predator- prey model with “ratio-dependent functional response, density dependent death rate of predator, self-diffusion terms corresponding to the random movement of the indi- viduals within two dimension, in addition with the influence of small amplitude heterogeneous perturbations to the linear intrinsic growth rates”.
Below we summarize the formulation in Sirohi et al. (2015) following their nota- tions, in order to show that the LV mechanism can be augmented by stochastic noise terms, still retaining non-trivial nonlinear behavior. We will use a modification of such a model while setting up the economic application in the next section. The nonlinear coupled partial differential equations representing the prey-predator interaction, are given as
∂u
∂t =u(1−u)− αuv
u+v + ∇2u≡ f(u, v)+ ∇2u,
∂v
∂t = βuv
u+v −γ v−δv2+d∇2v≡g(u, v)+ ∇2v (10.5)
whereu ≡u(x,y,t)andv≡v(x,y,t)denote the population densities of prey and predator respectively at a generic time point (t) and withinΩ ⊂R2with boundary
∂Ω in the real space. Initial conditions have been set to
u(x,y,0) >0, v(x,y,0) >0 ∀(x,y)∈Ω (10.6) The no-flux boundary conditions are represented in the following way,
∂u
∂ν =∂v
∂ν =0 (10.7)
∀(x,y)∈∂Ω and positive time points, whereν is the unit normal vector drawn outward on∂Ω, with scaler parametersα,β,γ andδ.
The magnitude of the parameters used in this model determine whether the Turing patterns will exist or not. In Sirohi et al. (2015), authors have consideredαanddas the bifurcation parameters for constructing the Turing bifurcation diagram. The Turing bifurcation diagram has been presented in αd-parametric plane (Fig.10.1; Sirohi et al. 2015). Value of the parameters in the given bifurcation diagram areβ =1, γ =0.6 andδ=0.1. In the present formulation,αanddcan be controlled to produce spatio-temporal patterns of different kinds. The curves shown in the figure are the Turing-bifurcation curve, temporal Hopf-bifurcation curve and temporal homoclinic bifurcation curve which have been marked as blue curve, red-dashed line and black- dotted line respectively. The equilibrium point E∗ destabilize atαh =2.01 which gives the Hopf-bifurcation curve. The condition for stability of the equilibrium point isα < αh. The region lying above the Turing bifurcation curve is the Turing instability region which is divided into two parts by the Hopf-bifurcation curve. Turing-Hopf domain with unstable temporal and spatio-temporal patterns lie in the region where α > αh. See Sirohi et al. (2015) for further numerical details.
Equation10.5 produces multiple interesting patterns which can be seen from Fig.10.1. Within the Turing domain, cold-spots and a mix of spots and stripes mate- rialize. Within the Turing-Hopf domain, a mix of spot-stripe, labyrinthine as well as chaotic patterns materialize. The patterns obtained in Fig.10.1have been marked with four different symbols depending on values ofαandd with the details in the caption.
The previous model (described in Eq.10.5) can be augmented by introducing uncorrelated multiplicative white noise terms. The new model is described as
∂u
∂t =u(1−u)−uα+vuv + ∇2u+σ1uξ1(t,x,y), (10.8)
∂v
∂t = βu+vuv −γ v−δv2+ ∇2v+σ2uξ2(t,x,y), (10.9) where ξ1(t,x,y) andξ2(t,x,y)are i.i.d. noise (a less strict definition would be temporally as well as spatially uncorrelated noise terms with normal distribution) with mean 0,
1.7 1.8 1.9 2 2.1 2.2 2.3 0
2 4 6 8 10 12 14 16 18
α
d
destabilization point
Fig. 10.1 Multiple types of spatial patterns can be observed for different parameter values within the Turing instability region in the parameter space. Four different colored symbols mark the relevant regions in the following way:◦cold-spot,mixture of spot-stripe, labyrinthine,∗chaotic and +interacting spiral. Turing bifurcation curve is shown byblue curve, Hopf-bifurcation curve by red dashed lineand temporal homoclinic bifurcation curve byblack dotted curve. Adapted from Anuj Kumar Sirohi Malay Banerjee and Anirban Chakraborti (2015)
E(ξ1(t,x,y))=E(ξ2(t,x,y))=0, (10.10) andσ1andσ2parameterizes the environmental effects.
Sirohi et al. (2015) conducts simulations of the system describes above and doc- uments that small magnitudes of noise intensities do not have any substantial effect on the spatiotemporal pattern formation apart from the fact that the time taken to reach the stationary pattern increases, which is expected. On the other hand, addition of small noise increases the irregularity within the non-stationary zone. They con- cluded noise and statistical interaction have vital roles in determining the distribution of species, even when environmental conditions are unfavourable.
To summarize the material discussed above, we have seen that simple LV systems generate many intricate patterns. In general not everything would be useful for eco- nomic applications. We borrow two well known insights from this literature. One, nonlinear dependence may lead to endogenous dynamics in the form of perpetual oscillation. That can, at least in principle, be useful to describe evolution of fluctuat- ing macroeconomic variables. Second, it is related to the idea that a dynamic system may not reach a constant equilibrium after all. This has been a point of discussion in multiple occasions among physicists and economists (Sinha et al.2010). Most of
the standard macroeconomic theory is driven by dynamical theories built around a constant equilibrium implying unless a shock hits the economy (productivity, mon- etary, fiscal policy etc.) it will not show any adjustment. However, it may seem to be somewhat unrealistic feature of such models.
Below we describe a model which is partly borrowed from the economic literature and partly depends on the LV type mechanisms described above. The idea is that given a shock process, an economic model describes evolution of macroeconomic quantities reasonably well. However, where that shock is coming from typically remains unanswered. We opine that in a multi-country context, there can a be a leader- follower relationship across countries that mimics the dynamics described above. The value addition of that approach is that the dynamics of the shock process can be totally endogenous and non-convergent to a constant equilibrium. Correspondingly, macro variables will also show a lag-lead structure as we describe below.