A novel approach for numeric study of 2D biological population model

Figures & data

Table 1. Comparison of numerical solution of Example 1 at $x=0.5,y=0.5$

	$t=0.1$	$t=0.2$	$t=0.3$	$t=0.4$	$t=0.5$
MCB-DQM	1.26282	1.14254	1.03381	0.93543	0.84641
IEFGM (Zhang et al., 2014)	1.2574	1.1382	1.0302	0.9325	0.8440
Exact	1.2628	1.1426	1.0339	0.9355	0.8465
CPU(s)	0.31	0.59	0.90	1.22	1.51

Table 2. The errors in Example 1 at different time levels $t\le 0.5$

$M=N$	Errors	$t=0.1$	$t=0.2$	$t=0.3$	$t=0.4$	$t=0.5$
13	$L_2$	2.689E-05	2.391E-05	2.141E-05	1.929E-05	1.747E-05
	$L_{\infty }$	1.052E-04	8.604E-05	7.037E-05	5.756E-05	4.709E-05
	CPU(s)	0.33	0.64	0.96	1.29	1.58
26	$L_2$	1.479E-05	1.183E-05	9.458E-06	7.568E-06	6.069E-06
	$L_{\infty }$	1.069E-04	8.716E-05	7.113E-05	5.806E-05	4.740E-05
	CPU(s)	2.31	4.62	6.81	9.14	11.14

Table 3. The errors in Example 1 for large time levels $5\le t\le 20$ with $h_x=h_y=1/12$

Errors	$t=5$	$t=10$	$t=15$	$t=20$
$L_2$	2.33E-07	1.57E-09	1.06E-11	7.14E-14
$L_{\infty }$	3.40E-07	2.30E-09	1.55E-11	1.04E-13
CPU(s)	15.17	30.31	45.34	61.43

Table 4. Rate of convergence (ROC) of Example 1

	$t=0.4$			$t=0.1$
$M=N$	$L_2$	ROC	CPU(s)	$L_2$	ROC	CPU(s)
5	1.17E-04		0.16	1.56E-04		0.04
10	2.85E-05	2.03	0.79	3.85E-05	2.02	0.20
15	1.22E-05	2.09	2.19	1.88E-05	1.76	0.60
20	8.23E-06	1.38	4.80	1.40E-05	1.02	1.29

Table 5. Errors in BPM given in Example 2 at $0\le t\le 1.0$ for different time steps and $h_x=h_y=0.1$

t	$▵t=0.001$			$▵t=0.0005$			$▵t=0.0001$
	$L_2$	$L_{\infty }$	CPU(s)	$L_2$	$L_{\infty }$	CPU(s)	$L_2$	$L_{\infty }$	CPU(s)
0.1	1.172E-04	3.689E-04	0.03	5.872E-05	1.827E-04	0.06	4.156E-05	6.096E-05	0.19
0.5	1.179E-04	3.691E-04	0.09	5.897E-05	1.828E-04	0.20	4.380E-05	6.265E-05	0.94
1.0	1.184E-04	3.694E-04	0.20	5.901E-05	1.829E-04	0.40	4.334E-05	6.164E-05	1.92

Table 6. Errors in BPM given in Example 2 at $0\le t\le 1.0$ with $▵t=0.0001$

$h_x=h_y$	Errors	$t=0.1$	$t=0.2$	$t=0.5$	$t=1.0$
0.03	$L_2$	6.43E-06	6.39E-06	6.36E-06	6.31E-06
	$L_{\infty }$	3.70E-05	3.69E-05	3.69E-05	3.69E-05
	CPU(s)	4.84	9.76	24.4	49.11
0.04	$L_2$	9.46E-06	9.53E-06	9.56E-06	9.39E-06
	$L_{\infty }$	3.65E-05	3.65E-05	3.65E-05	3.65E-05
	CPU(s)	2.23	4.48	11.24	22.52

Table 7. The errors in Example 2 for large time levels $5\le t\le 20$ with $h_x=h_y=1/12$

Errors	$t=5$	$t=10$	$t=15$	$t=20$
$L_2$	2.71E-05	2.50E-05	2.33E-05	2.19E-05
$L_{\infty }$	3.64E-05	3.65E-05	3.65E-05	3.66E-05
CPU(s)	12.14	30.22	45.67	60.94

Table 8. Rate of convergence of Example 2

	$t=1.0$			$t=0.5$
$M=N$	$L_2$	ROC	CPU(s)	$L_2$	ROC	CPU(s)
6	1.66E-04		0.36	1.67E-04		0.20
11	4.33E-05	2.21	1.90	4.38E-05	2.21	0.94
16	1.78E-05	2.38	5.45	1.80E-05	2.37	2.70
21	9.46E-06	2.32	12.00	9.53E-06	2.34	5.93
26	6.84E-06	1.52	22.33	6.34E-06	1.91	11.4

Table 9. Comparison of numerical solution of Example 3 at $x=0.5,y=0.5$

	$t=0.1$	$t=0.2$	$t=0.3$	$t=0.4$	$t=0.5$
MCB-DQM	0.5078	0.5170	0.5273	0.5379	0.5488
IEFGM (Zhang et al., 2014)	0.5069	0.5165	0.5268	0.5373	0.5481
Exact	0.5101	0.5204	0.5309	0.5416	0.5526
CPU(s)	0.31	0.61	0.94	1.20	1.48

Table 10. The errors in Example 3 for large time levels $2\le t\le 10$ with $h_x=h_y=1/12$

Errors	$t=2$	$t=5$	$t=8$	$t=10$
$L_2$	7.94E-03	1.50E-02	3.06E-02	2.07E-02
$L_{\infty }$	2.31E-03	4.10E-03	1.13E-02	9.34E-02
CPU(s)	5.87	14.84	23.79	29.60

Figure 1. The absolute errors BPM in Example 1 at different time levels $t\le 1$ with parameters $h_x=h_y=0.04,▵t=0.0001$.

Figure 2. The approximate solution of BPM in Example 1 at different time levels $t\le 1$ with parameters $h_x=h_y=0.04,▵t=0.0001$.

Figure 3. Contour and surface plot of absolute errors of $\mathit{\rho }(\mathbf{x},t)$ given in Example 2 for $t=0.1,0.5,1$ with parameters $h_x=h_y=0.05,▵t=0.0001$.

Figure 4. The behavior of $\mathit{\rho }(\mathbf{x},t)$ in Example 2 for $t=0.1,0.5,1$ with parameters $h_x=h_y=0.05,▵t=0.0001$.

Figure 5. Contour plot of absolute errors of $\mathit{\rho }(\mathbf{x},t)$ in Example 3 for $t=0.1,0.2,0.3,0.5$ with parameters $h_x=h_y=0.05,▵t=0.0001$.

Figure 6. Surface behavior of BPM as given in Example 3 for $t=0:1;0:2;0:3;0:5$ with parameters $h_x=h_y=0:05;▵t=0:0001$.

Zhang, L. W., Deng, Y. J., & Liew, K. M. (2014). An improved element-free Galerkin method for numerical modelling of bilological population problems. Engineering Analysis with Boundary Elements, 40, 181–188.

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