In this work, we studied two topics. The first theme is the effect of synapse plasticity of dynamics of neural system, and the second theme is the influence of Allee effect on population growth in an ecosystem. To do this, several computational models were used to check the changes in dynamics and obtain a phase diagram.
In the first topic, we consider an avalanche dynamics of neural network with synaptic plasticity. We compared the results of neural avalanche over several complex networks. On each network structure, we obtained artificial synapse strength that avalanche size distribution was transitions to a critical state that follows the power law, and obtained corresponding critical exponent. The critical exponents showed robustness to change of network structure. A critical exponent of neural avalanche size distribution and life time distribution did not change significantly when network structure changes.
In sparse random networks, although the number of connections was reduced, critical exponents were similar to those of a fully-connected network. And in scale-free networks and small-world networks, there were significant changes in critical parameters, but corresponding critical exponents did not. Next, we checked change in oscillation pattern of neural activity. Using these patterns, we classified three states of neural activity such as inactive, oscillation, and active.
In the second topic, in an ecosystem, we investigated the pattern of changes in population density when Allee effect is applied. Chaotic behavior was confirmed by adding the Allee effect to the logistic map. We obtained phase diagram showing extinction, convergence, divergence and chaotic behavior according to the growth rate and Allee threshold. When the Allee effect is applied, two independent trees appear in the bifurcation diagram representing the fixed point of the logistic map. Population growth rates and Allee thresholds cause these two trees to get closer or farther apart. Accordingly, various changes appear in the fixed point of the system. Finally, we investigated the spread of species when the Allee effect is applied in one dimensional discrete space. Here, according to the diffusion rate, it was confirmed whether the species succeeded in settle, and this result was extended with range of growth rate and Allee threshold.
The invasion velocity is limited by the rate of diffusion. Accordingly, we observed a pinning phenomenon in which invasion is restricted at a low diffusion rate and absorbing in which invasion is restricted at a relatively high diffusion rate. In absorbing regions, the invasion velocity is zero, i.e., even if the population moves, it cannot settle in the area. The pinning point at low diffusion rates increases as the population growth rate and Allee threshold increase. However, the absorbing point at high diffusion rates increases as the population growth rate increases, but decreases as the Allee threshold increases.