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Article Contents

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

• We study numerically a coagulation-fragmentation model derived by Niwa [17] and further elaborated by Degond et al. [5]. In [5] a unique equilibrium distribution of group sizes is shown to exist in both cases of continuous and discrete group size distributions. We provide a numerical investigation of these equilibria using three different methods to approximate the equilibrium: a recursive algorithm based on the work of Ma et. al. [12], a Newton method and the resolution of the time-dependent problem. All three schemes are validated by showing that they approximate the predicted small and large size asymptotic behaviour of the equilibrium accurately. The recursive algorithm is used to investigate the transition from discrete to continuous size distributions and the time evolution scheme is exploited to show uniform convergence to equilibrium in time and to determine convergence rates.

Mathematics Subject Classification: Primary: 45J05, 49M15, 82B99, 92D50.

 Citation:

• 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)

Table 1.  Starting with a uniform distribution 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 $L=100$, $h=0.01$, $\Delta t=1$ and $m_1 =1$. This approximation, $f_i(t)$, is evaluated at $t = 5,10,15,20$ and the equilibrium distribution is approximated via following the Euler scheme until $t=30$. The table shows the relative distances to the equilibrium, $\frac{\left|f_i^{\infty} - f_i(t)\right|}{f_i^{\infty} }$ for $t = 5,10,15,20,25$ and $i = 500,3500,6500,9500$

 Time $t$ $x=5$ $x=35$ $x=65$ $x=95$ $t=5$ $0.2772$ $9.2148$ $154.7531$ $2046.0000$ $t=10$ $0.0638$ $1.4009$ $10.8288$ $67.0145$ $t=15$ $0.0089$ $0.1976$ $1.0721$ $3.8651$ $t=20$ $0.0012$ $0.0260$ $0.1300$ $0.3832$ $t=25$ $0.0001$ $0.0030$ $0.0149$ $0.0423$

Table 2.  Starting with an exponential distribution 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 $L=100$, $h=0.01$, $\Delta t=0.5$ (smaller than in the previous case due to stabilisation problems for small sizes) and $m_1 =1$. This approximation, $f_i(t)$, is evaluated at $t = 5,10,15,20$ and the equilibrium distribution is approximated via following the Euler scheme until $t=30$. The table shows the relative distances to the equilibrium, $\frac{\left|f_i^{\infty} - f_i(t)\right|}{f_i^{\infty} }$ for $t = 5,10,15,20,25$ and $i = 500,3500,6500,9500$

 Time $t$ $x=5$ $x=35$ $x=65$ $x=95$ $t=5$ $0.05620$ $0.75370$ $0.98890$ $0.99980$ $t=10$ $0.00800$ $0.16010$ $0.50000$ $0.77500$ $t=15$ $0.00120$ $0.02670$ $0.11420$ $0.25710$ $t=20$ $0.00020$ $0.00430$ $0.02000$ $0.05220$ $t=25$ $0.00003$ $0.00004$ $0.00290$ $0.00790$
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