Numerical approximation of a coagulation-fragmentation model for animal group size statistics

Figure 1.  The equilibrium distribution is approximated by the Newton scheme (Section 5.2.3). In Fig. 1a, we take mass $m_1 =1$, grid size $h=0.01$ and the cut-off at $L=100$. The plot shows the generated distribution (blue solid line) in a log scale against the group sizes in a linear scale and presents the theoretically found large-size asymptotic behaviour (red dashed line) in a log scale for the sake of comparison. The group sizes are taken in a linear scale in order to illustrate the leading behaviour for large group sizes as a straight line. For Fig. 1b, the equilibrium distribution is approximated by the Newton scheme taking mass $m_1 =1$, grid size $h=0.0005$ and the cut-off at $L=5$. The plot shows the generated distribution (blue solid line) in a log scale and presents the theoretically found asymptotic small-size behaviour (red dashed line) in a log scale for the sake of comparison. The group sizes are taken in a log scale as well in order to illustrate the leading behaviour close to zero as a straight line

Figure 2.  The equilibrium distribution is approximated by simulating the time evolution of the distribution via the Euler scheme. Starting with a uniform distribution, the equilibrium, is reached at $T=30$ at the latest. In Fig. 2a, we take mass $m_1 =1$, grid size $h=0.01$ and the cut-off at $L=100$. The plot shows the generated distribution (blue solid line) in a log scale as a function of the group sizes in a linear scale and presents the theoretically found large-size asymptotic behaviour (red dashed line) in a log scale for the sake of comparison. The group sizes are taken in a linear scale in order to illustrate the leading behaviour for large group sizes as a straight line. For Fig. 2b, the equilibrium distribution is approximated by the Euler scheme taking mass $m_1 =1$, grid size $h=0.0005$ and the cut-off at $L=5$. The plot shows the generated distribution (blue solid line) in a log scale and presents the theoretically found asymptotic small-size behaviour (red dashed line) in a log scale for the sake of comparison. The group sizes are taken in a log scale as well in order to illustrate the leading behaviour close to zero as a straight line

Figure 3.  The equilibrium distribution is approximated by the recursive scheme (Section 5.1). In Fig. 3a, the mass is $m_1^h =1$, the grid size $h=1$ and the equilibrium sequence is computed till $L=100$. It shows the generated distribution (blue solid line) in a log scale and presents the theoretically found asymptotic behaviour (red dashed line) in a log scale (Eq. 6.3) for the sake of comparison. One can observe perfect agreement for large sizes. In Fig. 3b, exactly the same is done for grid size $h=0.01$. Again, one can observe that the generated distribution shows the predicted asymptotics

Figure 4.  Comparison of the equilibria for model D' ((5.5)-(5.6)) and model D ((2.15)-(2.18)). We take truncation $L=100$, grid size $h=0.01$ and mass $m_1 = m_1^h =1$. The equilibrium for model D' is generated by the Newton scheme (Section 5.2.3) and represented in a log scale by the solid blue line. The equilibrium for model D is generated by the recursive scheme (Section 5.1) and represented in a log scale by the dotted red line

Figure 5.  The equilibrium distribution is generated by the recursive scheme, for mass $m_1 =1$, taking grid size $h=1$, $h=0.1$ and $h=0.01$. The figure shows the generated distributions (solid lines) and the large-size asymptotic behaviour for model C ((2.19)-(2.23)) (dashed line) in a log scale (equation (6.2)). We have magnified the plot close to $x=200$

Figure 6.  The large-size behaviours of the discrete and continuous equilibrium distributions are compared, for mass $m_1 =1$ and fixed grid size $h=0.01$, close to $x=200$ (Fig. 6a), close to $x=1000$(Fig. 6b) and close to $x=2000$ (Fig. 6c). In each case, it shows the large-size asymptotic behaviour for model D ((2.15)-(2.18)) given by equation (6.3) (blue dotted line) and the large-size asymptotic behaviour for model C ((2.19)-(2.23)) given by equation (6.2) (red dashed line) in a log scale. Observe that for $x$ becoming greater, the difference between both graphs increases significantly

Figure 7.  In Fig. 7a we plot $\frac{1}{h} f_1^h$ for $h \in [5*10^{-5}, 1]$ in log-log scale (blue solid line) and the small-size asymptotics of the continuous model C ((2.19)-(2.23)) (red dashed line). For small $h$, the graphs illustrate the findings in (6.6). In Fig. 7b, the equilibrium sequence for model D ((2.15)-(2.18)) is generated as described in Section 5.1 taking mass $m_1^h =1$ and grid size $h = 5*10^{-5}$. The plot shows the distribution $(f_i^h)_{i \in \mathbb{N}} $ as a function of the group size in log-log scale (blue solid line) in the interval $[h,1]$ and the small-size asymptotics of the continuous model C (red dashed line). Both graphs tend to have the same slope for the sizes becoming smaller except for a slight divergence at the smallest group sizes

Figure 8.  Starting with a uniform distribution (Fig. 8a) and with an exponential distribution (Fig. 8b), the time-dependent solution of model C ((2.19)-(2.23)), $f(x,t)$, is approximated via the Euler scheme for model D' ((5.5)-(5.6)), taking $L=100$, $h=0.01$, $m_1 =1$ and $\Delta t=1$ for uniform initial and $\Delta t = 0.5$ for exponential initial (due to stability issues for small sizes). The approximation, $f_i(t)$, is evaluated at $t = 20,\dots, 29$ and the equilibrium distribution is approximated via following the Euler scheme until $t=30$. Calculating the relative distances to the equilibrium, $ \mu_i(t) = \left|f_i^{\infty} - f_i(t)\right|/f_i^{\infty} $, for $i = 500$ and $i=9500$ (representing $x=5$ and $x=95$), we estimate the exponential convergence rate $\delta_{x,t_2}$ ($\sim \delta_{x,t_1}$) for $t_1 = 20,\dots, 28$ and $t_2 = t_1 +1$ according to Eq. (6.8)