Evolutionary Dynamics: 1-Consumer + 2-Resources Small World
Mathematical Modeling + Simulation
DKWC
2021-08-21
1 Ecological Dynamics
The following mathematical ecological model simulation, variable notations and definitions are adapted to the paper the Wickman, Diehl, and Brännström (2019) in order not to get any confusion for readers, except
- consumer’s mortality rate is \(\lambda\) but not \(\mu\)
- resources up-taken affinities competed by consumer \(i\) for resource \(j\) is in matrix form \(a_{(i, j)}\) but in not scalar notation.
- the resource contribution function by \(j\) for gross consumer \(i\) growth is in matrix form \(C_{(i,j)}\) but in not scalar notation.
- The above matrix form settings are better for future work of Agent-based Simulation.
1.1 Model Assumpution
The Assumptions of this mathematical ecological model have fours
- the ecological Dynamics is to assume a homogeneous and closed system. Further research work will relax the assumption of homogeneity to heterogeneity late after.
- Total weights or total contributions are normalized as 1
- The symmetry property of \(G(X,Y) = G(Y,X)\) will occur at the equilibrium point.
- The assumption of monotonic property for \(C_{(i,j)}\) has a significant mathematical implication which can help to simplify computational programming and improve simulation’s efficiency :
Monotonic Properties for \(C_{(i,j)}\), if
1.2 Variable Definitions
2 N-Consumers + M Resources World
2.1 Consumer dynanmics: Dim(N, :)
2.2 Resource Dynanmics: Dim(:,M)
3 1-Consumer + 2-Resources Small World
3.1 Consumer dynanmics: Dim(1,:)
3.2 Resource Dynanmics: Dim(:,2)
3.3 subject to Constriants
4 Reparameterized 1-Consumer + 2-Resources Small World: Dim(1,2)
4.1 Define
\[\begin{align*} X &:= a_{(1,1)}R_{1}(t) \\ Y &:= a_{(1,2)}R_{2}(t) \end{align*}\]
4.2 Reparameterized the ODE Problem
4.3 State-space Representation
4.4 Assumption
- Assume \(r = K =1\) and \(G(X,Y)-\lambda = 0\)
4.5 Phase Portrait \(R_1(t)\) Vs \(R_2(t)\)
Phase Portrait Plot
4.6 Mathematical Implication
Because the symmetric property of \(G(X,Y) = G(Y,X)\) must occur at the equilibrium point, which implies that \[X(t \rightarrow \infty) = Y(t \rightarrow \infty)\]
\[\begin{align*} \therefore \frac{ X(t \rightarrow \infty)}{ Y(t \rightarrow \infty)} &= \left( \frac{a_{(1, 1)}}{a_{(1, 2)}}\right) = 1 \end{align*}\]
The above mathematical derivation shows that under a homogeneous 1-Consumer with 2-Resources ecological closed system if
the numbers of consumer is constant over time \(\frac{\mathrm{d} u_{1}(t)}{\mathrm{d} t} = 0\),
\(r = K =1\), and
the ratio of \(\left( \frac{a_{(1, 1)}}{a_{(1, 2)}}\right)= 1,\)
then the equilibrium resource point \((R^{*}_{1}(t), R^{*}_{2}(t))\) will reach at \((1,1)\)
This mathematical implication is inlined with the following Figure that published at the paper of Wickman, Diehl, and Brännström (2019) (Figure 6).
sourced from Wickman, Diehl, and Brännström (2019) (Figure 6)
Furthermore, the resource contribution function \(C_{(i, j)}\) is an endogenous control variable which is to govern the rate of reaching to the equilibrium resource point. All results in Figure 6 shown at Wickman, Diehl, and Brännström (2019) are subject to the curvature of function \(C_{(i, j)}\) with three scenario cases :
- monotonic increasing function,
- monotonic decreasing function, and
- hyperbolic/exponential function.
If the \(C_{(i, j)}\) is a hyperbolic/exponential function, this is the case of fast reaching to the ratio of \[\left( \frac{a_{(1, 1)}}{a_{(1, 2)}}\right) \rightarrow 1\], and then fast reaching to the equilibrium resource point \[(R_{1}(t\rightarrow \infty), R_{2}(t\rightarrow \infty))\rightarrow (1, 1).\]
Besides, the Phase Portrait Diagram is summarized the curvature of resource trade-off function between \(R_{1}(t)\) and \(R_{2}(t)\), where velocity arrows indicate both directions and managitudes that also inlined the arguments of Wickman, Diehl, and Brännström (2019) (Figure 6).
Finally, if \(r\) and \(K\) are not \(1\) but any other positive values (\(r\) and \(K \in(0,1]\)), all above mathematical implication mentioned for the small world dynamic Dim(1,2) are still valid; except it will reach to a new resource equilibrium \[(R_{1}(t\rightarrow \infty), R_{2}(t\rightarrow \infty))\rightarrow (K, K),\] where the renewal rate r is to govern the rate of exponentially decaying process in trading-off resources.
5 Small World: Heterogeneous environments
- Will be published later
6 Big Big World: N-Consumers + M Resources
- Will be published later