## 3 Numerical Method and Solutions
### The Effect of Parameters on Equilibrium Solutions
#### 3.3.1 Single Group Dynamics  
We start with a single group to isolate the impact of changing potentials and parameters on a group before introducing inter-group interactions. As previously discussed, the parameter \(\eta\) influences the diffusivity strength within the group, which competes with the force of the nonlocal term, \(K*u\), responsible for group aggregation. The interplay between these two terms determines the group's territory density and size. The top row of Figure 2 presents an illustrative example of how equilibrium solutions change when the diffusion strength, \(\eta\), is varied while keeping \(a\) and \(b\) fixed. This means that both the environmental and interaction potential strengths remain constant. As anticipated, the territory size increases with \(\eta\); for instance, in Figure 2c, when \(\eta=9\), the territory is more spread out compared to Figure 2a,b, where \(\eta=5\) and \(\eta=7\), respectively. The second row of Figure 2 shows a variation in the strength of the interaction potential. In Figure 2d-f, the parameter \(b\) is adjusted, while all other parameters remain fixed. It is important to note that the population densities are displayed relative to the environment, and as \(b\) increases, a clear aggregation of the population can be observed.  
The parameters of the potential also play a crucial role in influencing the aggregation of population densities. Figure 3 provides a visual representation of the solution for a single group using Gaussians with varying variances. It is evident that larger variances lead to a more spread-out population territory.  
The territory arrangement of the groups can also be influenced by environmental factors. Figure 4 illustrates the equilibrium population density of a group existing in a fragmented environment. This group is subject to an interacting potential denoted as \(K\), following a Gaussian distribution with \(\sigma=1\). The environmental potential, \(U(x)\), is composed of four Gaussians specified by \(U(x)=A[e^{-\frac{1}{2}(x^{2}+(y-5)^{2})}+e^{-\frac{1}{2}((x-5)^{2}+y^{2})}+e^ {-\frac{1}{2}(x^{2}+(y+5)^{2})}+\]  
Figure 2: The equilibrium solutions to System (2) with \(U(x)=Ae^{-0.05x^{2}-0.05y^{2}}\), where \(A\) is chosen so \(U\) integrates to \(1\), and \(K\) is a Gaussian with \(\sigma=1\). The simulations illustrated in the **top** row fix \(b=0.375\) and vary the value of \(\eta\). The **bottom** row fixes \(\eta=5\), and varies \(b\). In all simulations, the environment \(U\) is displayed in black and the population density in teal.  
\(e^{-\frac{1}{2}((x+5)^{2}+y^{2})}\)], and it is normalized to integrate to one. The upper row offers an bird's eye view of the territories, while the lower row depicts a three-dimensional view showcasing both the environmental potential and the population density. All simulations showcased in Figure 4 utilize the parameters \(\eta=3\) and \(b=0.5\). The two columns present distinct strengths of the environmental potential. Figure 4a,c illustrate different viewpoints of the same simulation, with a value of \(a=0.25\). Remarkably, in this instance, the environment is less attractive, leading to territory aggregation. As the environmental potency intensifies, as demonstrated in Figure 4b,d, the dominance of the environmental potential surpasses the aggregation strength within the group, ultimately leading to territorial segregation.