Plastic energy allocation toward life-history functions in a consumer-resource interaction

Abstract

Various environmental alterations resulting from the current global change compromise the persistence of species in their habitual environment. To cope with the obvious risk of extinction, plastic responses provide organisms with rapid acclimatization to new environments. The premise of plastic rescue has been theoretically studied from mathematical models in both deterministic and stochastic environments, focusing on analyzing the persistence and stability of the populations. Here, we evaluate this premise in the framework of a consumer-resource interaction considering the energy investment towards reproduction vs. maintenance as a plastic trait according to positive/negative variation of the available resource. A basic consumer-resource mathematical model is formulated based on the principle of biomass conversion that incorporates the energy allocation toward vital functions of the life-cycle of consumer individuals. Our mathematical approach is based on the impulsive differential equations at fixed moments considering two impulsive effects associated with the instants at which consumers obtain environmental information and when energy allocation strategy change occurs. From a preliminary analysis of the non-plastic temporal dynamics, namely when the energy allocation is constant over time and without experiencing changes concerning the variation of resources, both the persistence and stability of the consumer-resource dynamic are dependent on the energy allocation strategies belonging to a set termed stability range. We found that the plastic energy allocation can promote a stable dynamical pattern in the consumer-resource interaction depending on both the magnitude of the energy allocation change and the time lag between environmental sensibility instants and when the expression of the plastic trait occurs.

Appendices

Coefficients of functions $F_1$, $G_1$, and $H_1$.

Let be $F_1(X,Y,Z)=a_{200}X^2+a_{020}Y^2+a_{110}XY+a_{101}XZ+a_{011}YZ+a_{210}X^{2}Y+a_{201}X^{2}Z+a_{111}XYZ+a_{120}XY^{2}+a_{021}Y^{2}Z+a_{102}XZ^{2}+a_{012}YZ^{2}+a_{300}X^3$, $G_{1}(X,Y,Z)=b_{200}X^2+b_{020}Y^2+b_{002}Z^2+b_{110}XY+b_{101}XZ+b_{011}YZ+b_{210}X^{2}Y+b_{201}X^{2}Z+b_{111}XYZ+b_{120}XY^{2}+b_{021}Y^{2}Z+b_{102}XZ^{2}+b_{012}YZ^{2}+b_{300}X^3$, and $H_{1}(X,Y,Z)=c_{200}X^2+c_{020}Y^2+c_{110}XY+c_{101}XZ+c_{011}YZ+c_{210}X^{2}Y+c_{201}X^{2}Z+c_{111}XYZ+c_{120}XY^{2}+c_{021}Y^{2}Z+c_{102}XZ^{2}+c_{012}YZ^{2}+c_{300}X^3$ where

$$\begin{aligned}&a_{200}=\frac{(1 - \alpha )^3 {\tilde{\varphi }} {\tilde{\delta }}^3 [2 \alpha + (1 - \alpha ) {\tilde{\varphi }} {\tilde{\delta }}]}{\alpha ^2 + (1 - \alpha ) \alpha {\tilde{\delta }} + (1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2}, \\&a_{020}=\frac{(1 - \alpha )^3 {\tilde{b}}^2 {\tilde{\delta }} [\alpha +(1-\alpha ){\tilde{\delta }}]}{(1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}]}, \\&a_{110}=\frac{(1 - \alpha )^3 \alpha {\tilde{\varphi }} {\tilde{b}} {\tilde{\delta }}^2}{\alpha ^2 + (1 - \alpha ) \alpha {\tilde{\delta }} + (1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2},\\ {}&a_{101}=\frac{(1 - \alpha )^2 {\tilde{\varphi }} {\tilde{b}} (\alpha (-1 + {\tilde{\delta }}) - {\tilde{\delta }}) {\tilde{\delta }}^2 (-3 \alpha + (-1 + \alpha ) {\tilde{\varphi }} {\tilde{\delta }})}{\sqrt{\alpha +(1-\alpha ){\tilde{\delta }}} (\alpha ^2 + (1 - \alpha ) \alpha {\tilde{\delta }} + (1 - a)^2 {\tilde{\varphi }} {\tilde{\delta }}^2)}, \\&a_{011}=\frac{(1 - \alpha ) {\tilde{b}}^2 (\alpha ^3 + (1 - \alpha ) \alpha ^2 (2 + {\tilde{\varphi }}) {\tilde{\delta }} + (1 - \alpha )^2 \alpha {\tilde{\delta }}^2 - (1 - \alpha )^3 {\tilde{\varphi }} {\tilde{\delta }}^3)}{\sqrt{\alpha +(1-\alpha ){\tilde{\delta }}} (\alpha ^2 + (1 - \alpha ) \alpha {\tilde{\delta }} + (1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2)},\\ {}&a_{210}=-\frac{(1 - \alpha )^3 \alpha ^2 {\tilde{b}} {\tilde{\delta }}^2 (-\alpha ^2 -2 (1 - \alpha ) \alpha {\tilde{\delta }} + (1 - \alpha )^2 (-1 + {\tilde{\varphi }}) {\tilde{\delta }}^2)}{[\alpha +(1-\alpha ){\tilde{\delta }}] ((1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}])},\\&a_{201}=-\frac{\alpha ^2 {\tilde{\varphi }} {\tilde{b}} ({\tilde{\delta }} - \alpha {\tilde{\delta }})^3 ({\tilde{\delta }} - \alpha (1 + {\tilde{\delta }})}{\sqrt{\alpha +(1-\alpha ){\tilde{\delta }}} ((1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}])},\\ {}&a_{111}=\frac{(1 - \alpha )^2 \alpha {\tilde{b}}^2 {\tilde{\delta }} (\alpha ^3 + 2 (1 - \alpha ) \alpha ^2 {\tilde{\delta }} - (1 - \alpha )^2 \alpha (-1 + {\tilde{\varphi }}) {\tilde{\delta }}^2 + (1 - \alpha )^3 {\tilde{\varphi }} {\tilde{\delta }}^3)}{\sqrt{\alpha +(1-\alpha ){\tilde{\delta }}} (\alpha ^2 + (1 - \alpha ) \alpha {\tilde{\delta }} + (1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2)}, \\&a_{120}=-\frac{(1 - \alpha )^4 \alpha {\tilde{b}}^2 {\tilde{\delta }}^2 [\alpha +(1-\alpha ){\tilde{\delta }}]}{(1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}]},\\ {}&a_{021}=-\frac{\alpha ({\tilde{b}} - \alpha {\tilde{b}})^3 {\tilde{\delta }} [\alpha +(1-\alpha ){\tilde{\delta }}]^{3/2}}{(1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}]},\\&a_{102}=-\frac{\alpha ^2 {\tilde{\varphi }} {\tilde{b}} ({\tilde{\delta }} - \alpha {\tilde{\delta }})^3 ({\tilde{\delta }} - \alpha (1 + {\tilde{\delta }}))}{\sqrt{\alpha +(1-\alpha ){\tilde{\delta }}} ((1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}])}, \\ {}&a_{012}=\frac{(1 - \alpha )^4 \alpha {\tilde{\varphi }} {\tilde{b}}^3 {\tilde{\delta }}^3}{(1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}]}, \\&a_{300}=\frac{\alpha ^3 {\tilde{\varphi }} (1 - \alpha )^4 {\tilde{\delta }}^4}{[\alpha +(1-\alpha ){\tilde{\delta }}] ((1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}])},\\ {}&b_{200}=\frac{(1 -\alpha ) \alpha ^2 {\tilde{\varphi }} {\tilde{\delta }}^2 (\alpha ^2 -2 (-1 + \alpha ) \alpha {\tilde{\delta }} + (1 - \alpha )^2 (1 + 2 {\tilde{\varphi }}) {\tilde{\delta }}^2)}{{\tilde{b}} [\alpha +(1-\alpha ){\tilde{\delta }}] ((1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}])},\\&b_{020}=\frac{(1 - \alpha )^2 \alpha {\tilde{b}} {\tilde{\delta }} [\alpha +(1-\alpha ){\tilde{\delta }}]}{(1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}]},\\&b_{002}=-\alpha {\tilde{\varphi }} {\tilde{b}} {\tilde{\delta }},\\&b_{110}=\frac{(1 - \alpha )^4 \alpha {\tilde{\varphi }}^2 {\tilde{\delta }}^4}{-\alpha ^3 +2 (-1 + \alpha ) \alpha ^2 {\tilde{\delta }} - (1 - \alpha )^2 \alpha (1 + {\tilde{\varphi }}) {\tilde{\delta }}^2 - (1 - \alpha )^3 {\tilde{\varphi }} {\tilde{\delta }}^3},\\ {}&b_{101}=\frac{\alpha {\tilde{\varphi }} {\tilde{\delta }} (\alpha ^3 - 2 (-1 + \alpha ) \alpha ^2 {\tilde{\delta }} + (1 - \alpha )^2 \alpha (1 + 2 {\tilde{\varphi }}) {\tilde{\delta }}^2 - (1 - \alpha )^3 {\tilde{\varphi }} {\tilde{\delta }}^3)}{\sqrt{\alpha +(1-\alpha ){\tilde{\delta }}} ((1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}])},\\&b_{011}=\frac{\alpha {\tilde{b}} (\alpha ^3 - 2 (-1 + \alpha ) \alpha ^2 {\tilde{\delta }} - (1 - \alpha )^2 \alpha (-1 + {\tilde{\varphi }}) {\tilde{\delta }}^2 - (1 - \alpha )^3 {\tilde{\varphi }} (1 +{\tilde{\varphi }}) {\tilde{\delta }}^3)}{\sqrt{\alpha +(1-\alpha ){\tilde{\delta }}}(\alpha ^2 + (1 - \alpha ) \alpha {\tilde{\delta }} + (1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2)}, \\ {}&b_{210}=\frac{(1 - \alpha )^2 \alpha ^2 {\tilde{\delta }}^2 ((1 - \alpha )^4 {\tilde{\varphi }}^2 {\tilde{\delta }}^4 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}]^3)}{[\alpha +(1-\alpha ){\tilde{\delta }}]^2 ((1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}])},\\&b_{201}=-\frac{(1 - \alpha )^2 \alpha ^3 {\tilde{\varphi }} {\tilde{\delta }}^3 (\alpha ^2 - 2 (-1 + \alpha ) \alpha {\tilde{\delta }} + (1 - \alpha )^2 (1 + 2 {\tilde{\varphi }}) {\tilde{\delta }}^2)}{[\alpha +(1-\alpha ){\tilde{\delta }}]^{3/2} ((1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}])},\\&b_{111}=\frac{(1 - \alpha ) \alpha ^2 {\tilde{b}} {\tilde{\delta }} (\alpha [\alpha +(1-\alpha ){\tilde{\delta }}]^3 + {\tilde{\varphi }} ({\tilde{\delta }} - \alpha {\tilde{\delta }})^2 (\alpha ^2 - 2 (-1 + \alpha ) \alpha {\tilde{\delta }} + (1 - \alpha )^2 (1 + 2 {\tilde{\varphi }}) {\tilde{\delta }}^2)}{[\alpha +(1-\alpha ){\tilde{\delta }}]^{3/2} ((1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}])},\\&b_{120}=-\frac{(1 - \alpha )^3 \alpha ^2 {\tilde{b}} {\tilde{\delta }}^2 [\alpha +(1-\alpha ){\tilde{\delta }}]}{(1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}]},\\ {}&b_{021}=-\frac{(1 - \alpha )^2 \alpha ^2 {\tilde{b}}^2 {\tilde{\delta }} [\alpha +(1-\alpha ){\tilde{\delta }}]^{3/2}}{(1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}]},\\&b_{102}=-\frac{(1 - \alpha ) \alpha ^3 {\tilde{\varphi }} {\tilde{b}} {\tilde{\delta }}^2 (\alpha ^2 - 2 (-1 + \alpha ) \alpha {\tilde{\delta }} + (1 - \alpha )^2 (1 + {\tilde{\varphi }}) {\tilde{\delta }}^2)}{[\alpha +(1-\alpha ){\tilde{\delta }}] ((1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}])}, \\&b_{012}=\frac{(1 - \alpha )^2 \alpha ^2 {\tilde{\varphi }} {\tilde{b}}^2 {\tilde{\delta }}^2 (\alpha ^2 - 2 (-1 + \alpha ) \alpha {\tilde{\delta }} + (1 - \alpha )^2 (1 + {\tilde{\varphi }}) {\tilde{\delta }}^2)}{[\alpha +(1-\alpha ){\tilde{\delta }}] ((1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}])},\\&b_{300}=-\frac{(1 - \alpha )^5 \alpha ^3 {\tilde{\varphi }}^2 {\tilde{\delta }}^6}{{\tilde{b}} [\alpha +(1-\alpha ){\tilde{\delta }}]^2 ((1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}])},\\ {}&c_{200}= \frac{\alpha {\tilde{\varphi }} (\alpha + (-1 + \alpha ) {\tilde{\delta }}) ({\tilde{\delta }} - \alpha {\tilde{\delta }})^3}{{\tilde{b}} \sqrt{ \alpha + {\tilde{\delta }} - \alpha {\tilde{\delta }}}((1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}])},\\&c_{020}=-\frac{(1 - \alpha )^4 {\tilde{b}} {\tilde{\delta }}^2 \sqrt{\alpha +(1-\alpha ){\tilde{\delta }}}}{(1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}]}, \\ {}&c_{110}=\frac{(1 - \alpha )^2 {\tilde{\delta }} ((1 - \alpha )^3 {\tilde{\varphi }} {\tilde{\delta }}^3 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}]^2)}{\sqrt{\alpha +(1-\alpha ){\tilde{\delta }}} ((1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}])},\\&c_{101}=\frac{(1 - \alpha )^2 \alpha {\tilde{\varphi }} {\tilde{\delta }}^2 (\alpha + 2 (-1 + \alpha ) {\tilde{\delta }})}{\alpha ^2 + (1 - \alpha ) \alpha {\tilde{\delta }} + (1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2},\\&c_{011}=-\frac{(1 - \alpha ) {\tilde{b}} (-\alpha ^3 + (-1 + \alpha ) \alpha ^2 {\tilde{\delta }} - 2 (1 - \alpha )^3 {\tilde{\varphi }} {\tilde{\delta }}^3)}{\alpha ^2 + (1 - \alpha ) \alpha {\tilde{\delta }} + (1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2},\\&c_{210}=\frac{(1 - \alpha )^4 \alpha ^2 {\tilde{\delta }}^3 (-\alpha ^2 - 2 (1 - \alpha ) \alpha {\tilde{\delta }} + (1 - \alpha )^2 (-1 + {\tilde{\varphi }}) {\tilde{\delta }}^2)}{[\alpha +(1-\alpha ){\tilde{\delta }}]^{3/2} ((1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}])},\\&c_{201}=\frac{\alpha ^2 {\tilde{\varphi }} ({\tilde{\delta }} - \alpha {\tilde{\delta }})^4 ({\tilde{\delta }} - \alpha (1 + {\tilde{\delta }}))}{[\alpha +(1-\alpha ){\tilde{\delta }}] ((1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}])},\\&c_{111}=-\frac{(1 - \alpha )^3 \alpha {\tilde{b}} {\tilde{\delta }}^2 (-\alpha ^3 + 2 (-1 + \alpha ) \alpha ^2 {\tilde{\delta }} + (1 - \alpha )^2 \alpha (-1 + {\tilde{\varphi }}) {\tilde{\delta }}^2 - (1 - \alpha )^3 {\tilde{\varphi }} {\tilde{\delta }}^3)}{-\alpha ^3 + 2 (-1 + \alpha ) \alpha ^2 {\tilde{\delta }} - (1 - \alpha )^2 \alpha (1 + {\tilde{\varphi }}) {\tilde{\delta }}^2 - (1 - \alpha )^3 {\tilde{\varphi }} {\tilde{\delta }}^3},\\&c_{120}=\frac{(1 - \alpha )^5 \alpha {\tilde{b}} {\tilde{\delta }}^3 \sqrt{\alpha +(1-\alpha ){\tilde{\delta }}}}{(1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}]},\\&c_{021}=\frac{(1 - \alpha )^4 \alpha {\tilde{b}}^2 {\tilde{\delta }}^2 [\alpha +(1-\alpha ){\tilde{\delta }}]}{(1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}]},\\ {}&c_{102}=\frac{(1 - a)^4 \alpha ^2 {\tilde{\varphi }} {\tilde{b}} {\tilde{\delta }}^4}{\sqrt{ \alpha + {\tilde{\delta }} - \alpha {\tilde{\delta }}} (\alpha ^2 + (1 - \alpha ) \alpha {\tilde{\delta }} + (1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2)},\\ \\&c_{012}=-\frac{(1 - \alpha )^5 \alpha {\tilde{\varphi }} {\tilde{b}}^2 {\tilde{\delta }}^4}{\sqrt{\alpha +(1-\alpha ){\tilde{\delta }}} (\alpha ^2 + (1 - \alpha ) \alpha {\tilde{\delta }} + (1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2)},\\ {}&c_{300}=-\frac{(1 - \alpha )^5 \alpha ^3 {\tilde{\varphi }} {\tilde{\delta }}^5}{{\tilde{b}} [\alpha +(1-\alpha ){\tilde{\delta }}]^{3/2}((1 - \alpha )^2 {\tilde{\varphi }} {\tilde{\delta }}^2 + \alpha [\alpha +(1-\alpha ){\tilde{\delta }}])}, \end{aligned}$$

with ${\tilde{\varphi }}={\tilde{\varphi }}_{r}\alpha +{\tilde{\varphi }}_{m}(1-\alpha )$.

Coefficients associated to the value ${\mathcal {V}}$

$a_{0}=-2{\tilde{\delta }}^8{\tilde{\varphi }}_m$, $a_1=-2 {\tilde{\delta }}^7 (-1 + (-1 + 15 {\tilde{\delta }}) {\tilde{\varphi }}_m - {\tilde{\delta }} {\tilde{\varphi }}_r)$, $a_2=-{\tilde{\delta }}^6 (5 + {\tilde{\delta }}^2 (210 {\tilde{\varphi }}_m + 3 {\tilde{\varphi }}_m^3 + 2 {\tilde{\varphi }}_m^4 - 28 {\tilde{\varphi }}_r) - 2 {\tilde{\delta }} (13 + 12 {\tilde{\varphi }}_m + {\tilde{\varphi }}_m^2 - {\tilde{\varphi }}_r))$, $a_3={\tilde{\delta }}^5 (-4 + 2 {\tilde{\delta }} (26 - 5 {\tilde{\varphi }}_m + {\tilde{\varphi }}_m^2) - {\tilde{\delta }}^2 (156 + 27 {\tilde{\varphi }}_m^2 + 5 {\tilde{\varphi }}_m^3 + {\tilde{\varphi }}_m (134 - 4 {\tilde{\varphi }}_r) - 22 {\tilde{\varphi }}_r) +{\tilde{\delta }}^3 (910 {\tilde{\varphi }}_m + 28 {\tilde{\varphi }}_m^4 + {\tilde{\varphi }}_m^3 (39 - 8 {\tilde{\varphi }}_r) - 182 {\tilde{\varphi }}_r - 9 {\tilde{\varphi }}_m^2 {\tilde{\varphi }}_r))$, $a_4=-{\tilde{\delta }}^4 (1 + {\tilde{\delta }} (-17 + 9 {\tilde{\varphi }}_m) + {\tilde{\delta }}^2 (21 {\tilde{\varphi }}_m^2 + 2 {\tilde{\varphi }}_m^3 + 10 (25 + {\tilde{\varphi }}_r) - 4 {\tilde{\varphi }}_m (31 + {\tilde{\varphi }}_r)) + {\tilde{\delta }}^3 (-52 {\tilde{\varphi }}_m^3 + 8 {\tilde{\varphi }}_m^4 + {\tilde{\varphi }}_m^2 (-167 + 15 {\tilde{\varphi }}_r) + {\tilde{\varphi }}_m (-464 + 50 {\tilde{\varphi }}_r) - 2 (286 - 56 {\tilde{\varphi }}_r + {\tilde{\varphi }}_r^2)) + {\tilde{\delta }}^4 (182 {\tilde{\varphi }}_m^4 - 728 {\tilde{\varphi }}_r + 12 {\tilde{\varphi }}_m^2 (-9 + {\tilde{\varphi }}_r) {\tilde{\varphi }}_r - 26 {\tilde{\varphi }}_m^3 (-9 + 4 {\tilde{\varphi }}_r) + 3 {\tilde{\varphi }}_m (910 + 3 {\tilde{\varphi }}_r^2)))$, $a_5={\tilde{\delta }}^4 (-23 - 4 {\tilde{\varphi }}_m + {\tilde{\delta }} (19 + 123 {\tilde{\varphi }}_m + 10 {\tilde{\varphi }}_m^2 - 9 {\tilde{\varphi }}_r) - 2 {\tilde{\delta }}^2 (-370 + 3 {\tilde{\varphi }}_m^3 + 4 {\tilde{\varphi }}_m^4 - 57 {\tilde{\varphi }}_r - {\tilde{\varphi }}_r^2 + {\tilde{\varphi }}_m^2 (-49 + 3 {\tilde{\varphi }}_r) +{\tilde{\varphi }}_m (338 + 19 {\tilde{\varphi }}_r)) + {\tilde{\delta }}^3 (-1430 + 88 {\tilde{\varphi }}_m^4 + 352 {\tilde{\varphi }}_r - 23 {\tilde{\varphi }}_r^2 - 2 {\tilde{\varphi }}_m^3 (125 + 16 {\tilde{\varphi }}_r) + 3 {\tilde{\varphi }}_m^2 (-209 + 47 {\tilde{\varphi }}_r) + {\tilde{\varphi }}_m (-1122 + 284 {\tilde{\varphi }}_r - 15 {\tilde{\varphi }}_r^2)) + {\tilde{\delta }}^4 (728 {\tilde{\varphi }}_m^4 + 18 {\tilde{\varphi }}_m^2 {\tilde{\varphi }}_r (-33 + 8 {\tilde{\varphi }}_r) - 78 {\tilde{\varphi }}_m^3 (-11 + 8 {\tilde{\varphi }}_r) - {\tilde{\varphi }}_r (2002 + 3 {\tilde{\varphi }}_r^2) + {\tilde{\varphi }}_m (6006 + 99 {\tilde{\varphi }}_r^2 - 8 {\tilde{\varphi }}_r^3)))$, $a_6=-{\tilde{\delta }}^3 (17 + {\tilde{\varphi }}_m + {\tilde{\delta }}^2 (284 + 106 {\tilde{\varphi }}_m^2 + 32 {\tilde{\varphi }}_m^3 + {\tilde{\varphi }}_m (665 - 20 {\tilde{\varphi }}_r) - 114 {\tilde{\varphi }}_r) - {\tilde{\delta }} (203 + 51 {\tilde{\varphi }}_m + 7 {\tilde{\varphi }}_m^2 - 4 {\tilde{\varphi }}_r) + {\tilde{\delta }}^3 (1515 - 80 {\tilde{\varphi }}_m^4 + 562 {\tilde{\varphi }}_r + 17 {\tilde{\varphi }}_r^2 + 3 {\tilde{\varphi }}_m^2 (91 + 8 {\tilde{\varphi }}_r) + 2 {\tilde{\varphi }}_m^3 (-71 + 16 {\tilde{\varphi }}_r) + 2 {\tilde{\varphi }}_m (-1077 - 79 {\tilde{\varphi }}_r + 3 {\tilde{\varphi }}_r^2)) + {\tilde{\delta }}^4 (-2574 + 440 {\tilde{\varphi }}_m^4 + 770 {\tilde{\varphi }}_r - 119 {\tilde{\varphi }}_r^2 + 5 {\tilde{\varphi }}_r^3 - 20 {\tilde{\varphi }}_m^3 (37 + 16 {\tilde{\varphi }}_r) + {\tilde{\varphi }}_m^2 (-1595 + 609 {\tilde{\varphi }}_r + 48 {\tilde{\varphi }}_r^2) - 2 {\tilde{\varphi }}_m (1012 - 485 {\tilde{\varphi }}_r + 63 {\tilde{\varphi }}_r^2)) + {\tilde{\delta }}^5 (2002 {\tilde{\varphi }}_m^4 + 396 {\tilde{\varphi }}_m^2 {\tilde{\varphi }}_r (-5 + 2 {\tilde{\varphi }}_r) - 143 {\tilde{\varphi }}_m^3 (-15 + 16 {\tilde{\varphi }}_r) + 2 {\tilde{\varphi }}_r (-2002 - 15 {\tilde{\varphi }}_r^2 + {\tilde{\varphi }}_r^3) - 11 {\tilde{\varphi }}_m (-910 - 45 {\tilde{\varphi }}_r^2 + 8 {\tilde{\varphi }}_r^3)))$, $a_7={\tilde{\delta }}^2 (-3 - {\tilde{\delta }}^2 (708 + 80 {\tilde{\varphi }}_m^2 + 12 {\tilde{\varphi }}_m^3 + {\tilde{\varphi }}_m (233 - 14 {\tilde{\varphi }}_r) - 47 {\tilde{\varphi }}_r) + {\tilde{\delta }} (95 - 5 {\tilde{\varphi }}_m + {\tilde{\varphi }}_m^2 - {\tilde{\varphi }}_r) + {\tilde{\delta }}^3 (864 + 256 {\tilde{\varphi }}_m^3 + {\tilde{\varphi }}_m (1980 - 192 {\tilde{\varphi }}_r) + {\tilde{\varphi }}_m^2 (472 - 96 {\tilde{\varphi }}_r) - 551 {\tilde{\varphi }}_r + 10 {\tilde{\varphi }}_r^2) - 2 {\tilde{\delta }}^4 (-1140 + 180 {\tilde{\varphi }}_m^4 + {\tilde{\varphi }}_m^3 (339 - 144 {\tilde{\varphi }}_r) - 796 {\tilde{\varphi }}_r - 31 {\tilde{\varphi }}_r^2 +{\tilde{\varphi }}_r^3 + {\tilde{\varphi }}_m (2238 + 194 {\tilde{\varphi }}_r + 15 {\tilde{\varphi }}_r^2) + {\tilde{\varphi }}_m^2 (-259 - 201 {\tilde{\varphi }}_r + 24 {\tilde{\varphi }}_r^2)) + {\tilde{\delta }}^6 (4004 {\tilde{\varphi }}_m^4 + 165 {\tilde{\varphi }}_m^2 {\tilde{\varphi }}_r (-27 + 16 {\tilde{\varphi }}_r) - 143 {\tilde{\varphi }}_m^3 (-27 + 40 {\tilde{\varphi }}_r) - 55 {\tilde{\varphi }}_m (-234 - 27 {\tilde{\varphi }}_r^2 + 8 {\tilde{\varphi }}_r^3) + {\tilde{\varphi }}_r (-6006 - 135 {\tilde{\varphi }}_r^2 + 20 {\tilde{\varphi }}_r^3)) + {\tilde{\delta }}^5 (-3432 + 1320 {\tilde{\varphi }}_m^4 + 1254 {\tilde{\varphi }}_r - 366 {\tilde{\varphi }}_r^2 + 37 {\tilde{\varphi }}_r^3 - 15 {\tilde{\varphi }}_m^3 (101 + 96 {\tilde{\varphi }}_r) + 3 {\tilde{\varphi }}_m^2 (-968 + 537 {\tilde{\varphi }}_r + 144 {\tilde{\varphi }}_r^2) - {\tilde{\varphi }}_m (2838 - 2220 {\tilde{\varphi }}_r + 483 {\tilde{\varphi }}_r^2 + 32 {\tilde{\varphi }}_r^3)))$, $a_8=-{\tilde{\delta }}^2 (-2 + 9 {\tilde{\varphi }}_m + {\tilde{\delta }} (237 + 23 {\tilde{\varphi }}_m^2 + 6 {\tilde{\varphi }}_r - 2 {\tilde{\varphi }}_m (37 + {\tilde{\varphi }}_r)) - {\tilde{\delta }}^2 (1398 + 84 {\tilde{\varphi }}_m^3 + {\tilde{\varphi }}_m (571 - 146 {\tilde{\varphi }}_r) - 186 {\tilde{\varphi }}_r + 7 {\tilde{\varphi }}_r^2 - 4 {\tilde{\varphi }}_m^2 (-85 + 9 {\tilde{\varphi }}_r)) + {\tilde{\delta }}^3 (1414 + 896 {\tilde{\varphi }}_m^3 - 1429 {\tilde{\varphi }}_r + 86 {\tilde{\varphi }}_r^2 - 168 {\tilde{\varphi }}_m^2 (-7 + 4 {\tilde{\varphi }}_r) + {\tilde{\varphi }}_m (3678 - 752 {\tilde{\varphi }}_r + 96 {\tilde{\varphi }}_r^2)) - 2 {\tilde{\delta }}^4 (480 {\tilde{\varphi }}_m^4 + {\tilde{\varphi }}_m^3 (846 - 576 {\tilde{\varphi }}_r) + {\tilde{\varphi }}_m^2 (-365 - 816 {\tilde{\varphi }}_r + 192 {\tilde{\varphi }}_r^2) + {\tilde{\varphi }}_m (3192 + 324 {\tilde{\varphi }}_r + 186 {\tilde{\varphi }}_r^2 - 16 {\tilde{\varphi }}_r^3) - 2 (651 + 721 {\tilde{\varphi }}_r + 33 {\tilde{\varphi }}_r^2 + 3 {\tilde{\varphi }}_r^3)) + 6 {\tilde{\delta }}^6 (1001 {\tilde{\varphi }}_m^4 - 858 {\tilde{\varphi }}_m^3 (-1 + 2 {\tilde{\varphi }}_r) + 198 {\tilde{\varphi }}_m^2 {\tilde{\varphi }}_r (-6 + 5 {\tilde{\varphi }}_r) - 55 {\tilde{\varphi }}_m (-39 - 9 {\tilde{\varphi }}_r^2 + 4 {\tilde{\varphi }}_r^3) + {\tilde{\varphi }}_r (-1144 - 60 {\tilde{\varphi }}_r^2 + 15 {\tilde{\varphi }}_r^3)) + 2 {\tilde{\delta }}^5 (-1716 + 1320 {\tilde{\varphi }}_m^4 + 792 {\tilde{\varphi }}_r - 372 {\tilde{\varphi }}_r^2 + 62 {\tilde{\varphi }}_r^3 + 4 {\tilde{\varphi }}_r^4 - 60 {\tilde{\varphi }}_m^3 (19 + 32 {\tilde{\varphi }}_r) + 3 {\tilde{\varphi }}_m^2 (-649 + 489 {\tilde{\varphi }}_r + 288 {\tilde{\varphi }}_r^2) - 2 {\tilde{\varphi }}_m (792 - 897 {\tilde{\varphi }}_r + 282 {\tilde{\varphi }}_r^2 + 64 {\tilde{\varphi }}_r^3)))$, $a_9={\tilde{\delta }} (-4 + {\tilde{\delta }} (19 + 61 {\tilde{\varphi }}_m - 4 {\tilde{\varphi }}_m^2 - 9 {\tilde{\varphi }}_r) + {\tilde{\delta }}^2 (355 + 101 {\tilde{\varphi }}_m^2 + 68 {\tilde{\varphi }}_r + {\tilde{\varphi }}_r^2 - {\tilde{\varphi }}_m (235 + 44 {\tilde{\varphi }}_r)) - {\tilde{\delta }}^3 (252 {\tilde{\varphi }}_m^3 - 8 {\tilde{\varphi }}_m^2 (-94 + 27 {\tilde{\varphi }}_r) + 11 (158 - 35 {\tilde{\varphi }}_r + 6 {\tilde{\varphi }}_r^2) +{\tilde{\varphi }}_m (869 - 534 {\tilde{\varphi }}_r + 36 {\tilde{\varphi }}_r^2)) + {\tilde{\delta }}^4 (1414 + 1792 {\tilde{\varphi }}_m^3 - 2249 {\tilde{\varphi }}_r + 290 {\tilde{\varphi }}_r^2 - 32 {\tilde{\varphi }}_r^3 - 28 {\tilde{\varphi }}_m^2 (-65 + 72 {\tilde{\varphi }}_r) + 2 {\tilde{\varphi }}_m (2249 - 800 {\tilde{\varphi }}_r + 288 {\tilde{\varphi }}_r^2)) - 4 {\tilde{\delta }}^5 (-570 + 420 {\tilde{\varphi }}_m^4 + {\tilde{\varphi }}_m^3 (651 - 672 {\tilde{\varphi }}_r) - 875 {\tilde{\varphi }}_r - 48 {\tilde{\varphi }}_r^2 - 28 {\tilde{\varphi }}_r^3 + 2 {\tilde{\varphi }}_r^4 + 7 {\tilde{\varphi }}_m^2 (-29 - 123 {\tilde{\varphi }}_r + 48 {\tilde{\varphi }}_r^2) - 7 {\tilde{\varphi }}_m (-228 - 29 {\tilde{\varphi }}_r - 45 {\tilde{\varphi }}_r^2 + 8 {\tilde{\varphi }}_r^3)) + 2 {\tilde{\delta }}^7 (3432 {\tilde{\varphi }}_m^4 + 594 {\tilde{\varphi }}_m^2 {\tilde{\varphi }}_r (-7 + 8 {\tilde{\varphi }}_r) - 858 {\tilde{\varphi }}_m^3 (-3 + 8 {\tilde{\varphi }}_r) + {\tilde{\varphi }}_m (5005 + 2079 {\tilde{\varphi }}_r^2 - 1320 {\tilde{\varphi }}_r^3) + 3 {\tilde{\varphi }}_r (-1001 - 105 {\tilde{\varphi }}_r^2 + 40 {\tilde{\varphi }}_r^3)) + 2 {\tilde{\delta }}^6 (-1287 + 1848 {\tilde{\varphi }}_m^4 + 792 {\tilde{\varphi }}_r - 525 {\tilde{\varphi }}_r^2 + 126 {\tilde{\varphi }}_r^3 + 28 {\tilde{\varphi }}_r^4 - 42 {\tilde{\varphi }}_m^3 (31 + 80 {\tilde{\varphi }}_r) + 3 {\tilde{\varphi }}_m^2 (-649 + 651 {\tilde{\varphi }}_r + 672 {\tilde{\varphi }}_r^2) - {\tilde{\varphi }}_m (1419 - 2100 {\tilde{\varphi }}_r + 903 {\tilde{\varphi }}_r^2 + 448 {\tilde{\varphi }}_r^3)))$, $a_{10}=-{\tilde{\delta }} (-4 (3 + {\tilde{\varphi }}_m) + {\tilde{\delta }} (36 - 16 {\tilde{\varphi }}_m^2 - 52 {\tilde{\varphi }}_r + 2 {\tilde{\varphi }}_m (69 + 4 {\tilde{\varphi }}_r)) + {\tilde{\delta }}^2 (355 + 195 {\tilde{\varphi }}_m^2 + 167 {\tilde{\varphi }}_r + 21 {\tilde{\varphi }}_r^2 - 2 {\tilde{\varphi }}_m (167 + 79 {\tilde{\varphi }}_r)) - {\tilde{\delta }}^3 (1398 + 420 {\tilde{\varphi }}_m^3 + {\tilde{\varphi }}_m^2 (970 - 540 {\tilde{\varphi }}_r) - 484 {\tilde{\varphi }}_r + 201 {\tilde{\varphi }}_r^2 - 12 {\tilde{\varphi }}_r^3 + {\tilde{\varphi }}_m (869 - 970 {\tilde{\varphi }}_r + 180 {\tilde{\varphi }}_r^2)) + {\tilde{\delta }}^4 (864 + 2240 {\tilde{\varphi }}_m^3 - 2249 {\tilde{\varphi }}_r + 510 {\tilde{\varphi }}_r^2 - 160 {\tilde{\varphi }}_r^3 - 140 {\tilde{\varphi }}_m^2 (-13 + 24 {\tilde{\varphi }}_r) + 6 {\tilde{\varphi }}_m (613 - 340 {\tilde{\varphi }}_r + 240 {\tilde{\varphi }}_r^2)) + {\tilde{\delta }}^7 (6006 {\tilde{\varphi }}_m^4 + 792 {\tilde{\varphi }}_m^2 {\tilde{\varphi }}_r (-9 + 14 {\tilde{\varphi }}_r) - 429 {\tilde{\varphi }}_m^3 (-9 + 32 {\tilde{\varphi }}_r) - 462 {\tilde{\varphi }}_m (-13 - 9 {\tilde{\varphi }}_r^2 + 8 {\tilde{\varphi }}_r^3) + 28 {\tilde{\varphi }}_r (-143 - 27 {\tilde{\varphi }}_r^2 + 15 {\tilde{\varphi }}_r^3)) + {\tilde{\delta }}^5 (1515 - 2016 {\tilde{\varphi }}_m^4 + 2884 {\tilde{\varphi }}_r + 214 {\tilde{\varphi }}_r^2 + 308 {\tilde{\varphi }}_r^3 - 48 {\tilde{\varphi }}_r^4 + 84 {\tilde{\varphi }}_m^3 (-31 + 48 {\tilde{\varphi }}_r) + {\tilde{\varphi }}_m^2 (730 + 4368 {\tilde{\varphi }}_r - 2688 {\tilde{\varphi }}_r^2) + 4 {\tilde{\varphi }}_m (-1119 - 203 {\tilde{\varphi }}_r - 546 {\tilde{\varphi }}_r^2 + 168 {\tilde{\varphi }}_r^3)) + 2 {\tilde{\delta }}^6 (-715 + 1848 {\tilde{\varphi }}_m^4 + 627 {\tilde{\varphi }}_r - 525 {\tilde{\varphi }}_r^2 + 175 {\tilde{\varphi }}_r^3 + 84 {\tilde{\varphi }}_r^4 - 12 {\tilde{\varphi }}_m^3 (95 + 336 {\tilde{\varphi }}_r) + 3 {\tilde{\varphi }}_m^2 (-484 + 651 {\tilde{\varphi }}_r + 1008 {\tilde{\varphi }}_r^2) - 2 {\tilde{\varphi }}_m (506 - 897 {\tilde{\varphi }}_r + 525 {\tilde{\varphi }}_r^2 + 448 {\tilde{\varphi }}_r^3)))$, $a_{11}={\tilde{\delta }} (4 (-3 - 2 {\tilde{\varphi }}_m + {\tilde{\varphi }}_r) + {\tilde{\delta }} (19 - 24 {\tilde{\varphi }}_m^2 - 86 {\tilde{\varphi }}_r - 4 {\tilde{\varphi }}_r^2 + 6 {\tilde{\varphi }}_m (23 + 4 {\tilde{\varphi }}_r)) + {\tilde{\delta }}^2 (237 + 195 {\tilde{\varphi }}_m^2 + 167 {\tilde{\varphi }}_r + 58 {\tilde{\varphi }}_r^2 - {\tilde{\varphi }}_m (235 + 232 {\tilde{\varphi }}_r)) + {\tilde{\delta }}^4 (284 + 1792 {\tilde{\varphi }}_m^3 - 1429 {\tilde{\varphi }}_r + 510 {\tilde{\varphi }}_r^2 - 320 {\tilde{\varphi }}_r^3 - 168 {\tilde{\varphi }}_m^2 (-7 + 20 {\tilde{\varphi }}_r) + 20 {\tilde{\varphi }}_m (99 - 80 {\tilde{\varphi }}_r + 96 {\tilde{\varphi }}_r^2)) - {\tilde{\delta }}^3 (708 + 420 {\tilde{\varphi }}_m^3 + {\tilde{\varphi }}_m^2 (752 - 720 {\tilde{\varphi }}_r) - 385 {\tilde{\varphi }}_r + 284 {\tilde{\varphi }}_r^2 - 48 {\tilde{\varphi }}_r^3 + {\tilde{\varphi }}_m (571 - 970 {\tilde{\varphi }}_r + 360 {\tilde{\varphi }}_r^2)) - 2 {\tilde{\delta }}^5 (-370 + 840 {\tilde{\varphi }}_m^4 + {\tilde{\varphi }}_m^3 (846 - 2016 {\tilde{\varphi }}_r) - 796 {\tilde{\varphi }}_r - 96 {\tilde{\varphi }}_r^2 - 210 {\tilde{\varphi }}_r^3 + 60 {\tilde{\varphi }}_r^4 + 7 {\tilde{\varphi }}_m^2 (-37 - 246 {\tilde{\varphi }}_r + 240 {\tilde{\varphi }}_r^2) + {\tilde{\varphi }}_m (1077 + 324 {\tilde{\varphi }}_r + 1092 {\tilde{\varphi }}_r^2 - 560 {\tilde{\varphi }}_r^3)) + {\tilde{\delta }}^7 (4004 {\tilde{\varphi }}_m^4 - 429 {\tilde{\varphi }}_m^3 (-5 + 24 {\tilde{\varphi }}_r) + 297 {\tilde{\varphi }}_m^2 {\tilde{\varphi }}_r (-15 + 32 {\tilde{\varphi }}_r) + 14 {\tilde{\varphi }}_r (-143 - 45 {\tilde{\varphi }}_r^2 + 36 {\tilde{\varphi }}_r^3) - 6 {\tilde{\varphi }}_m (-455 - 495 {\tilde{\varphi }}_r^2 + 616 {\tilde{\varphi }}_r^3)) + {\tilde{\delta }}^6 (-572 + 2640 {\tilde{\varphi }}_m^4 + 770 {\tilde{\varphi }}_r - 744 {\tilde{\varphi }}_r^2 + 350 {\tilde{\varphi }}_r^3 + 280 {\tilde{\varphi }}_r^4 - 15 {\tilde{\varphi }}_m^3 (101 + 448 {\tilde{\varphi }}_r) + {\tilde{\varphi }}_m^2 (-1595 + 2934 {\tilde{\varphi }}_r + 6048 {\tilde{\varphi }}_r^2) - 2 {\tilde{\varphi }}_m (561 - 1110 {\tilde{\varphi }}_r + 903 {\tilde{\varphi }}_r^2 + 1120 {\tilde{\varphi }}_r^3)))$, $a_{12}=-{\tilde{\delta }} (-4 (1 + {\tilde{\varphi }}_m - {\tilde{\varphi }}_r) - {\tilde{\delta }} (2 + 16 {\tilde{\varphi }}_m^2 + 52 {\tilde{\varphi }}_r + 8 {\tilde{\varphi }}_r^2 - {\tilde{\varphi }}_m (61 + 24 {\tilde{\varphi }}_r)) + {\tilde{\delta }}^2 (95 + 101 {\tilde{\varphi }}_m^2 + 68 {\tilde{\varphi }}_r + 58 {\tilde{\varphi }}_r^2 - 2 {\tilde{\varphi }}_m (37 + 79 {\tilde{\varphi }}_r)) - {\tilde{\delta }}^3 (203 + 252 {\tilde{\varphi }}_m^3 + {\tilde{\varphi }}_m^2 (340 - 540 {\tilde{\varphi }}_r) - 186 {\tilde{\varphi }}_r + 201 {\tilde{\varphi }}_r^2 - 72 {\tilde{\varphi }}_r^3 + {\tilde{\varphi }}_m (233 - 534 {\tilde{\varphi }}_r + 360 {\tilde{\varphi }}_r^2)) + {\tilde{\delta }}^4 (19 + 896 {\tilde{\varphi }}_m^3 - 551 {\tilde{\varphi }}_r + 290 {\tilde{\varphi }}_r^2 - 320 {\tilde{\varphi }}_r^3 -8 {\tilde{\varphi }}_m^2 (-59 + 252 {\tilde{\varphi }}_r) + {\tilde{\varphi }}_m (665 - 752 {\tilde{\varphi }}_r + 1440 {\tilde{\varphi }}_r^2)) + {\tilde{\delta }}^7 (2002 {\tilde{\varphi }}_m^4 + 1980 {\tilde{\varphi }}_m^2 {\tilde{\varphi }}_r (-1 + 3 {\tilde{\varphi }}_r) - 286 {\tilde{\varphi }}_m^3 (-3 + 20 {\tilde{\varphi }}_r) + {\tilde{\varphi }}_m (910 + 1485 {\tilde{\varphi }}_r^2 - 2640 {\tilde{\varphi }}_r^3) + 4 {\tilde{\varphi }}_r (-182 - 90 {\tilde{\varphi }}_r^2 + 105 {\tilde{\varphi }}_r^3)) + {\tilde{\delta }}^5 (250 - 960 {\tilde{\varphi }}_m^4 + 562 {\tilde{\varphi }}_r + 132 {\tilde{\varphi }}_r^2 + 308 {\tilde{\varphi }}_r^3 - 160 {\tilde{\varphi }}_r^4 + 6 {\tilde{\varphi }}_m^3 (-113 + 448 {\tilde{\varphi }}_r) - 3 {\tilde{\varphi }}_m^2 (-91 - 544 {\tilde{\varphi }}_r + 896 {\tilde{\varphi }}_r^2) + 4 {\tilde{\varphi }}_m (-169 - 97 {\tilde{\varphi }}_r - 315 {\tilde{\varphi }}_r^2 + 280 {\tilde{\varphi }}_r^3)) + {\tilde{\delta }}^6 (-156 + 1320 {\tilde{\varphi }}_m^4 + 352 {\tilde{\varphi }}_r - 366 {\tilde{\varphi }}_r^2 + 252 {\tilde{\varphi }}_r^3 + 280 {\tilde{\varphi }}_r^4 - 20 {\tilde{\varphi }}_m^3 (37 + 192 {\tilde{\varphi }}_r) + 3 {\tilde{\varphi }}_m^2 (-209 + 537 {\tilde{\varphi }}_r + 1344 {\tilde{\varphi }}_r^2) - 2 {\tilde{\varphi }}_m (232 - 485 {\tilde{\varphi }}_r + 564 {\tilde{\varphi }}_r^2 + 896 {\tilde{\varphi }}_r^3)))$, $a_{13}={\tilde{\delta }}^2 (-3 - 4 {\tilde{\varphi }}_m^2 - 9 {\tilde{\varphi }}_r - 4 {\tilde{\varphi }}_r^2 + {\tilde{\varphi }}_m (9 + 8 {\tilde{\varphi }}_r) + {\tilde{\delta }} (17 + 23 {\tilde{\varphi }}_m^2 + 6 {\tilde{\varphi }}_r + 21 {\tilde{\varphi }}_r^2 - {\tilde{\varphi }}_m (5 + 44 {\tilde{\varphi }}_r)) - {\tilde{\delta }}^2 (23 + 84 {\tilde{\varphi }}_m^3 + {\tilde{\varphi }}_m^2 (80 - 216 {\tilde{\varphi }}_r) - 47 {\tilde{\varphi }}_r + 66 {\tilde{\varphi }}_r^2 - 48 {\tilde{\varphi }}_r^3 + {\tilde{\varphi }}_m (51 - 146 {\tilde{\varphi }}_r + 180 {\tilde{\varphi }}_r^2)) + {\tilde{\delta }}^3 (-17 + 256 {\tilde{\varphi }}_m^3 + {\tilde{\varphi }}_m^2 (106 - 672 {\tilde{\varphi }}_r) - 114 {\tilde{\varphi }}_r + 86 {\tilde{\varphi }}_r^2 - 160 {\tilde{\varphi }}_r^3 + 3 {\tilde{\varphi }}_m (41 - 64 {\tilde{\varphi }}_r + 192 {\tilde{\varphi }}_r^2)) - 2 {\tilde{\delta }}^4 (-26 + 180 {\tilde{\varphi }}_m^4 + {\tilde{\varphi }}_m^3 (71 - 576 {\tilde{\varphi }}_r) - 57 {\tilde{\varphi }}_r - 31 {\tilde{\varphi }}_r^2 - 56 {\tilde{\varphi }}_r^3 + 60 {\tilde{\varphi }}_r^4 + {\tilde{\varphi }}_m^2 (-49 - 201 {\tilde{\varphi }}_r + 672 {\tilde{\varphi }}_r^2) + {\tilde{\varphi }}_m (62 + 79 {\tilde{\varphi }}_r + 186 {\tilde{\varphi }}_r^2 - 336 {\tilde{\varphi }}_r^3)) + {\tilde{\delta }}^6 (728 {\tilde{\varphi }}_m^4 + 66 {\tilde{\varphi }}_m^2 {\tilde{\varphi }}_r (-9 + 40 {\tilde{\varphi }}_r) - 26 {\tilde{\varphi }}_m^3 (-9 + 88 {\tilde{\varphi }}_r) - 15 {\tilde{\varphi }}_m (-14 - 33 {\tilde{\varphi }}_r^2 + 88 {\tilde{\varphi }}_r^3) + {\tilde{\varphi }}_r (-182 - 135 {\tilde{\varphi }}_r^2 + 240 {\tilde{\varphi }}_r^3)) + {\tilde{\delta }}^5 (-26 + 440 {\tilde{\varphi }}_m^4 + 112 {\tilde{\varphi }}_r - 119 {\tilde{\varphi }}_r^2 + 124 {\tilde{\varphi }}_r^3 + 168 {\tilde{\varphi }}_r^4 - 10 {\tilde{\varphi }}_m^3 (25 + 144 {\tilde{\varphi }}_r) + {\tilde{\varphi }}_m^2 (-167 + 609 {\tilde{\varphi }}_r + 1728 {\tilde{\varphi }}_r^2) - {\tilde{\varphi }}_m (134 - 284 {\tilde{\varphi }}_r + 483 {\tilde{\varphi }}_r^2 + 896 {\tilde{\varphi }}_r^3)))$, $a_{14}=-{\tilde{\delta }}^3 ({\tilde{\varphi }}_m + {\tilde{\varphi }}_m^2 - 2 {\tilde{\varphi }}_m {\tilde{\varphi }}_r + (-1 + {\tilde{\varphi }}_r) {\tilde{\varphi }}_r + {\tilde{\delta }} (1 - 12 {\tilde{\varphi }}_m^3 + 4 {\tilde{\varphi }}_r - 7 {\tilde{\varphi }}_r^2 + 12 {\tilde{\varphi }}_r^3 + {\tilde{\varphi }}_m^2 (-7 + 36 {\tilde{\varphi }}_r) - 2 {\tilde{\varphi }}_m (2 - 7 {\tilde{\varphi }}_r + 18 {\tilde{\varphi }}_r^2)) + {\tilde{\delta }}^2 (-4 + 32 {\tilde{\varphi }}_m^3 + {\tilde{\varphi }}_m^2 (10 - 96 {\tilde{\varphi }}_r) - 9 {\tilde{\varphi }}_r + 10 {\tilde{\varphi }}_r^2 - 32 {\tilde{\varphi }}_r^3 +{\tilde{\varphi }}_m (9 - 20 {\tilde{\varphi }}_r + 96 {\tilde{\varphi }}_r^2)) + {\tilde{\delta }}^5 (182 {\tilde{\varphi }}_m^4 + {\tilde{\varphi }}_m^3 (39 - 624 {\tilde{\varphi }}_r) - 28 {\tilde{\varphi }}_r - 30 {\tilde{\varphi }}_r^3 + 90 {\tilde{\varphi }}_r^4 + 36 {\tilde{\varphi }}_m^2 {\tilde{\varphi }}_r (-3 + 22 {\tilde{\varphi }}_r) + {\tilde{\varphi }}_m (30 + 99 {\tilde{\varphi }}_r^2 - 440 {\tilde{\varphi }}_r^3)) + {\tilde{\delta }}^3 (5 - 80 {\tilde{\varphi }}_m^4 + 10 {\tilde{\varphi }}_r + 17 {\tilde{\varphi }}_r^2 + 12 {\tilde{\varphi }}_r^3 - 48 {\tilde{\varphi }}_r^4 + 6 {\tilde{\varphi }}_m^3 (-1 + 48 {\tilde{\varphi }}_r) + {\tilde{\varphi }}_m^2 (21 + 24 {\tilde{\varphi }}_r - 384 {\tilde{\varphi }}_r^2) + 2 {\tilde{\varphi }}_m (-5 - 19 {\tilde{\varphi }}_r - 15 {\tilde{\varphi }}_r^2 + 112 {\tilde{\varphi }}_r^3)) + {\tilde{\delta }}^4 (-2 + 88 {\tilde{\varphi }}_m^4 + 22 {\tilde{\varphi }}_r - 23 {\tilde{\varphi }}_r^2 + 37 {\tilde{\varphi }}_r^3 + 56 {\tilde{\varphi }}_r^4 - 4 {\tilde{\varphi }}_m^3 (13 + 80 {\tilde{\varphi }}_r) + 3 {\tilde{\varphi }}_m^2 (-9 + 47 {\tilde{\varphi }}_r + 144 {\tilde{\varphi }}_r^2) - 2 {\tilde{\varphi }}_m (12 - 25 {\tilde{\varphi }}_r + 63 {\tilde{\varphi }}_r^2 + 128 {\tilde{\varphi }}_r^3)))$, $a_{15}={\tilde{\delta }}^6 ({\tilde{\varphi }}_m - {\tilde{\varphi }}_r) ({\tilde{\delta }}^2 (2 + 28 {\tilde{\varphi }}_m^3 + {\tilde{\varphi }}_m^2 (3 - 76 {\tilde{\varphi }}_r) + 3 {\tilde{\varphi }}_r^2 - 20 {\tilde{\varphi }}_r^3 + 2 {\tilde{\varphi }}_m {\tilde{\varphi }}_r (-3 + 34 {\tilde{\varphi }}_r)) + 2 (-4 {\tilde{\varphi }}_m^3 + {\tilde{\varphi }}_m^2 (1 + 12 {\tilde{\varphi }}_r) + {\tilde{\varphi }}_m (1 - 2 {\tilde{\varphi }}_r - 12 {\tilde{\varphi }}_r^2) + {\tilde{\varphi }}_r (-1 + {\tilde{\varphi }}_r + 4 {\tilde{\varphi }}_r^2)) + {\tilde{\delta }} (-2 + 8 {\tilde{\varphi }}_m^3 + 2 {\tilde{\varphi }}_r - 5 {\tilde{\varphi }}_r^2 - 8 {\tilde{\varphi }}_r^3 - {\tilde{\varphi }}_m^2 (5 + 24 {\tilde{\varphi }}_r) + 2 {\tilde{\varphi }}_m (-1 + 5 {\tilde{\varphi }}_r + 12 {\tilde{\varphi }}_r^2)))$, $a_{16}=-2 {\tilde{\delta }}^8 ({\tilde{\varphi }}_m - {\tilde{\varphi }}_r)^4$.

$b_{0}=-8 {\tilde{\delta }}^7 {\tilde{\varphi }}_m^3$, $b_{1}=8 {\tilde{\delta }}^6 {\tilde{\varphi }}_m^2 (-2 + (-1 + 12 {\tilde{\delta }}) {\tilde{\varphi }}_m - 3 {\tilde{\delta }} {\tilde{\varphi }}_r)$, $b_2=-8 {\tilde{\delta }}^5 {\tilde{\varphi }}_m (1 + {\tilde{\delta }} (-7 + 66 {\tilde{\delta }}) {\tilde{\varphi }}_m^2 + 4 {\tilde{\delta }} {\tilde{\varphi }}_r + 3 {\tilde{\delta }}^2 {\tilde{\varphi }}_r^2 + {\tilde{\varphi }}_m (4 - 33 {\tilde{\delta }}^2 {\tilde{\varphi }}_r + {\tilde{\delta }} (-20 + 3 {\tilde{\varphi }}_r)))$, $b_{3}=8 {\tilde{\delta }}^4 ({\tilde{\delta }} (-8 - 19 {\tilde{\delta }} + 220 {\tilde{\delta }}^2) {\tilde{\varphi }}_m^3 - {\tilde{\delta }} {\tilde{\varphi }}_r (1 + {\tilde{\delta }} {\tilde{\varphi }}_r)^2 + {\tilde{\varphi }}_m^2 (-2 + 28 {\tilde{\delta }} + 18 {\tilde{\delta }}^2 (-5 + {\tilde{\varphi }}_r) - 165 {\tilde{\delta }}^3 {\tilde{\varphi }}_r) + {\tilde{\varphi }}_m (-3 - 8 {\tilde{\delta }} (-1 + {\tilde{\varphi }}_r) - 3 {\tilde{\delta }}^2 (-12 + {\tilde{\varphi }}_r) {\tilde{\varphi }}_r + 30 {\tilde{\delta }}^3 {\tilde{\varphi }}_r^2))$, $b_{4}=-8 {\tilde{\delta }}^3 ({\tilde{\delta }} (4 - 64 {\tilde{\delta }} - 21 {\tilde{\delta }}^2 + 495 {\tilde{\delta }}^3) {\tilde{\varphi }}_m^3 + {\tilde{\delta }} {\tilde{\varphi }}_r (3 + {\tilde{\delta }}^2 (-16 + {\tilde{\varphi }}_r) {\tilde{\varphi }}_r - 9 {\tilde{\delta }}^3 {\tilde{\varphi }}_r^2 + {\tilde{\delta }} (-7 + 4 {\tilde{\varphi }}_r)) + {\tilde{\varphi }}_m (3 + {\tilde{\delta }}^2 (28 - 48 {\tilde{\varphi }}_r) + 135 {\tilde{\delta }}^4 {\tilde{\varphi }}_r^2 + {\tilde{\delta }} (-17 + 4 {\tilde{\varphi }}_r) -3 {\tilde{\delta }}^3 {\tilde{\varphi }}_r (-48 + 5 {\tilde{\varphi }}_r)) + {\tilde{\delta }} {\tilde{\varphi }}_m^2 (8 - 495 {\tilde{\delta }}^3 {\tilde{\varphi }}_r + 8 {\tilde{\delta }} (11 + 3 {\tilde{\varphi }}_r) + 3 {\tilde{\delta }}^2 (-80 + 13 {\tilde{\varphi }}_r)))$, $b_{5}=8 {\tilde{\delta }}^2 (2 {\tilde{\delta }}^2 (14 - 112 {\tilde{\delta }} + 3 {\tilde{\delta }}^2 + 396 {\tilde{\delta }}^3) {\tilde{\varphi }}_m^3 - {\tilde{\delta }} {\tilde{\varphi }}_r (3 + {\tilde{\delta }}^2 (21 - 20 {\tilde{\varphi }}_r) + 2 {\tilde{\delta }} (-7 + {\tilde{\varphi }}_r) - 4 {\tilde{\delta }}^3 (-14 + {\tilde{\varphi }}_r) {\tilde{\varphi }}_r + 36 {\tilde{\delta }}^4 {\tilde{\varphi }}_r^2) - 2 {\tilde{\delta }} {\tilde{\varphi }}_m^2 (12 + 495 {\tilde{\delta }}^4 {\tilde{\varphi }}_r - 84 {\tilde{\delta }}^2 (1 + {\tilde{\varphi }}_r) - 6 {\tilde{\delta }}^3 (-35 + 2 {\tilde{\varphi }}_r) + {\tilde{\delta }} (-44 + 6 {\tilde{\varphi }}_r)) + {\tilde{\varphi }}_m (-1 + 2 {\tilde{\delta }} - 24 {\tilde{\delta }}^4 (-14 + {\tilde{\varphi }}_r) {\tilde{\varphi }}_r + 360 {\tilde{\delta }}^5 {\tilde{\varphi }}_r^2 - {\tilde{\delta }}^2 (43 + 20 {\tilde{\varphi }}_r) - 8 {\tilde{\delta }}^3 (-7 + 16 {\tilde{\varphi }}_r + 3 {\tilde{\varphi }}_r^2)))$, $b_{6}=-8 {\tilde{\delta }}^2 (14 {\tilde{\delta }}^2 (6 - 32 {\tilde{\delta }} + 3 {\tilde{\delta }}^2 + 66 {\tilde{\delta }}^3) {\tilde{\varphi }}_m^3 - 2 {\tilde{\varphi }}_m^2 (-4 + 60 {\tilde{\delta }} + 693 {\tilde{\delta }}^5 {\tilde{\varphi }}_r + 21 {\tilde{\delta }}^4 (12 + {\tilde{\varphi }}_r) + 4 {\tilde{\delta }}^2 (-31 + 9 {\tilde{\varphi }}_r) - 28 {\tilde{\delta }}^3 (4 + 9 {\tilde{\varphi }}_r)) + {\tilde{\varphi }}_r (1 + {\tilde{\delta }} + 4 {\tilde{\delta }}^4 (-28 + {\tilde{\varphi }}_r) {\tilde{\varphi }}_r - 84 {\tilde{\delta }}^5 {\tilde{\varphi }}_r^2 + {\tilde{\delta }}^2 (29 + 12 {\tilde{\varphi }}_r) + {\tilde{\delta }}^3 (-35 + 44 {\tilde{\varphi }}_r + 8 {\tilde{\varphi }}_r^2)) + {\tilde{\varphi }}_m (19 + 504 {\tilde{\delta }}^4 {\tilde{\varphi }}_r + 630 {\tilde{\delta }}^5 {\tilde{\varphi }}_r^2 + {\tilde{\delta }} (-19 + 48 {\tilde{\varphi }}_r) + {\tilde{\delta }}^2 (-65 - 156 {\tilde{\varphi }}_r + 12 {\tilde{\varphi }}_r^2) - 2 {\tilde{\delta }}^3 (-35 + 104 {\tilde{\varphi }}_r + 72 {\tilde{\varphi }}_r^2)))$, $b_{7}=8 {\tilde{\delta }} (2 {\tilde{\delta }}^3 (70 - 280 {\tilde{\delta }} + 21 {\tilde{\delta }}^2 + 396 {\tilde{\delta }}^3) {\tilde{\varphi }}_m^3 - 2 {\tilde{\delta }} {\tilde{\varphi }}_m^2 (-16 + 120 {\tilde{\delta }} + 693 {\tilde{\delta }}^5 {\tilde{\varphi }}_r + 42 {\tilde{\delta }}^4 (5 + {\tilde{\varphi }}_r) + 10 {\tilde{\delta }}^2 (-17 + 9 {\tilde{\varphi }}_r) - 28 {\tilde{\delta }}^3 (4 + 15 {\tilde{\varphi }}_r)) - {\tilde{\delta }} {\tilde{\varphi }}_r (20 + 126 {\tilde{\delta }}^5 {\tilde{\varphi }}_r^2 + 4 {\tilde{\delta }}^4 {\tilde{\varphi }}_r (35 + {\tilde{\varphi }}_r) + 6 {\tilde{\delta }} (-3 + 4 {\tilde{\varphi }}_r) + {\tilde{\delta }}^2 (-36 - 66 {\tilde{\varphi }}_r + 4 {\tilde{\varphi }}_r^2) - 5 {\tilde{\delta }}^3 (-7 + 12 {\tilde{\varphi }}_r + 8 {\tilde{\varphi }}_r^2)) + {\tilde{\varphi }}_m (-16 + {\tilde{\delta }} (62 - 16 {\tilde{\varphi }}_r) + 756 {\tilde{\delta }}^6 {\tilde{\varphi }}_r^2 + 42 {\tilde{\delta }}^5 {\tilde{\varphi }}_r (12 + {\tilde{\varphi }}_r) + 12 {\tilde{\delta }}^2 (-3 + 16 {\tilde{\varphi }}_r) + 5 {\tilde{\delta }}^3 (-13 - 68 {\tilde{\varphi }}_r + 12 {\tilde{\varphi }}_r^2) - 8 {\tilde{\delta }}^4 (-7 + 30 {\tilde{\varphi }}_r + 45 {\tilde{\varphi }}_r^2)))$, $b_{8}=-8 ({\tilde{\delta }}^4 (140 - 448 {\tilde{\delta }} + 6 {\tilde{\delta }}^2 + 495 {\tilde{\delta }}^3) {\tilde{\varphi }}_m^3 - 2 {\tilde{\delta }}^2 {\tilde{\varphi }}_m^2 (-24 + 120 {\tilde{\delta }} + 495 {\tilde{\delta }}^5 {\tilde{\varphi }}_r - 84 {\tilde{\delta }}^3 (1 + 5 {\tilde{\varphi }}_r) + 3 {\tilde{\delta }}^4 (40 + 7 {\tilde{\varphi }}_r) + 4 {\tilde{\delta }}^2 (-31 + 30 {\tilde{\varphi }}_r)) + {\tilde{\delta }} {\tilde{\varphi }}_r (16 + {\tilde{\delta }}^2 (18 - 72 {\tilde{\varphi }}_r) - 126 {\tilde{\delta }}^6 {\tilde{\varphi }}_r^2 - 2 {\tilde{\delta }}^5 {\tilde{\varphi }}_r (56 + 5 {\tilde{\varphi }}_r) + {\tilde{\delta }} (-42 + 8 {\tilde{\varphi }}_r) + {\tilde{\delta }}^3 (29 + 104 {\tilde{\varphi }}_r - 16 {\tilde{\varphi }}_r^2) + {\tilde{\delta }}^4 (-21 + 60 {\tilde{\varphi }}_r + 80 {\tilde{\varphi }}_r^2)) + {\tilde{\varphi }}_m (4 - 32 {\tilde{\delta }} + {\tilde{\delta }}^2 (62 - 48 {\tilde{\varphi }}_r) + 630 {\tilde{\delta }}^7 {\tilde{\varphi }}_r^2 + 42 {\tilde{\delta }}^6 {\tilde{\varphi }}_r (8 + {\tilde{\varphi }}_r) + {\tilde{\delta }}^3 (-19 + 288 {\tilde{\varphi }}_r) + {\tilde{\delta }}^4 (-43 - 340 {\tilde{\varphi }}_r + 120 {\tilde{\varphi }}_r^2) - 4 {\tilde{\delta }}^5 (-7 + 52 {\tilde{\varphi }}_r + 120 {\tilde{\varphi }}_r^2)))$, $b_{9}=8 ({\tilde{\delta }}^4 (84 - 224 {\tilde{\delta }} - 21 {\tilde{\delta }}^2 + 220 {\tilde{\delta }}^3) {\tilde{\varphi }}_m^3 + {\tilde{\delta }}^2 {\tilde{\varphi }}_m^2 (32 - 120 {\tilde{\delta }} + {\tilde{\delta }}^2 (88 - 180 {\tilde{\varphi }}_r) - 495 {\tilde{\delta }}^5 {\tilde{\varphi }}_r + 6 {\tilde{\delta }}^4 (-15 + 4 {\tilde{\varphi }}_r) + 8 {\tilde{\delta }}^3 (11 + 63 {\tilde{\varphi }}_r)) + {\tilde{\varphi }}_r (-4 + 16 {\tilde{\delta }} + {\tilde{\delta }}^3 (1 - 72 {\tilde{\varphi }}_r) - 84 {\tilde{\delta }}^7 {\tilde{\varphi }}_r^2 - 4 {\tilde{\delta }}^6 {\tilde{\varphi }}_r (14 + {\tilde{\varphi }}_r) + 4 {\tilde{\delta }}^2 (-5 + 4 {\tilde{\varphi }}_r) + {\tilde{\delta }}^4 (14 + 66 {\tilde{\varphi }}_r - 24 {\tilde{\varphi }}_r^2) + {\tilde{\delta }}^5 (-7 + 44 {\tilde{\varphi }}_r + 80 {\tilde{\varphi }}_r^2)) + {\tilde{\varphi }}_m (4 - 16 {\tilde{\delta }} + {\tilde{\delta }}^2 (19 - 48 {\tilde{\varphi }}_r) + 144 {\tilde{\delta }}^6 {\tilde{\varphi }}_r + 360 {\tilde{\delta }}^7 {\tilde{\varphi }}_r^2 + 2 {\tilde{\delta }}^3 (1 + 96 {\tilde{\varphi }}_r) - 8 {\tilde{\delta }}^5 (-1 + 16 {\tilde{\varphi }}_r + 45 {\tilde{\varphi }}_r^2) + {\tilde{\delta }}^4 (-17 - 156 {\tilde{\varphi }}_r + 120 {\tilde{\varphi }}_r^2)))$, $b_{10}=8 {\tilde{\delta }}^2 ((-4 + 8 {\tilde{\delta }} - 5 {\tilde{\delta }}^3) (2 - 2 {\tilde{\delta }} + {\tilde{\delta }}^2 (7 {\tilde{\varphi }}_m - 4 {\tilde{\varphi }}_r)) ({\tilde{\varphi }}_m - {\tilde{\varphi }}_r)^2 - 2 {\tilde{\delta }}^3 (-4 + {\tilde{\delta }} + 5 {\tilde{\delta }}^2) ({\tilde{\varphi }}_m - {\tilde{\varphi }}_r)^3 - (-1 + {\tilde{\delta }}) (21 {\tilde{\delta }}^4 {\tilde{\varphi }}_m^3 + {\tilde{\delta }}^2 {\tilde{\varphi }}_m^2 (8 - 10 {\tilde{\delta }} - 45 {\tilde{\delta }}^2 {\tilde{\varphi }}_r) - {\tilde{\varphi }}_r (1 - 2 {\tilde{\delta }} + {\tilde{\delta }}^2 (1 - 4 {\tilde{\varphi }}_r) + 6 {\tilde{\delta }}^3 {\tilde{\varphi }}_r + 6 {\tilde{\delta }}^4 {\tilde{\varphi }}_r^2) + {\tilde{\varphi }}_m (1 - 2 {\tilde{\delta }} + {\tilde{\delta }}^2 (1 - 12 {\tilde{\varphi }}_r) + 16 {\tilde{\delta }}^3 {\tilde{\varphi }}_r + 30 {\tilde{\delta }}^4 {\tilde{\varphi }}_r^2)))$, $b_{11}=8 {\tilde{\delta }}^4 ({\tilde{\varphi }}_m - {\tilde{\varphi }}_r)^2 (-2 + 4 {\tilde{\varphi }}_m + 3 {\tilde{\delta }}^3 (4 {\tilde{\varphi }}_m - 3 {\tilde{\varphi }}_r) - 4 {\tilde{\varphi }}_r + {\tilde{\delta }}^2 (-2 - 7 {\tilde{\varphi }}_m + 4 {\tilde{\varphi }}_r) + {\tilde{\delta }} (4 - 8 {\tilde{\varphi }}_m + 8 {\tilde{\varphi }}_r))$, $b_{12}=-8 (-1 + {\tilde{\delta }}) {\tilde{\delta }}^6 ({\tilde{\varphi }}_m - {\tilde{\varphi }}_r)^3$.

Simulation procedures