Finding Implicit and Explicit Solutions to Initial Value Problems

Parallel Implicit Solution of 3-D Navier-Stokes Equations

Ü. Gülçat , V. Ünal , in Parallel Computational Fluid Dynamics 2002, 2003

Parallel implicit solution of Navier-Stokes equations based on two fractional steps in time and Finite Element discretization in space is presented. The accuracy of the scheme is second order in both time and space domains. Large time step sizes, with CFL numbers much larger than unity, are used. The Domain Decomposition Technique is implemented for parallel solution of the problem with matching and non-overlapping sub domains. Lid-driven flow in a cubic cavity with Reynolds number of 400 and 1000 is selected as a test case. The solution domain is divided into 2, 4 and 6 sub-domains. Time accurate solutions are obtained with time steps 5 times the step size of a stable explicit method. Super-linear speed-up is achieved with the modified Domain Decomposition Technique.

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First-Order Ordinary Differential Equations

Martha L. Abell , James P. Braselton , in Differential Equations with Mathematica (Fourth Edition), 2016

We will devote a considerable amount of time in this text to developing explicit, implicit, numerical, and graphical solutions of differential equations. In this chapter we introduce frequently encountered forms of first-order ordinary differential equations and methods to construct explicit, numerical, and graphical solutions of them. Several of the equations along with the methods of solution discussed here will be used in subsequent chapters of the text.

Charles Emile Picard (July 24, 1856, Paris, France-December 11, 1941, Paris, France) According to O'Connor, "Picard and his wife had three children, a daughter and two sons, who were all killed in World War I. His grandsons were wounded and captured in World War II." In regards to Picard's teaching, the famous French mathematician Jacques Salomon Hadamard (1865–1963) wrote in Picard's obituary "A striking feature of Picard's scientific personality was the perfection of his teaching, one of the most marvelous, if not the most marvelous that I have known."See texts like [6], [7], or [3].

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Introduction to differential equations

Henry J. Ricardo , in A Modern Introduction to Differential Equations (Third Edition), 2021

Implicit solutions

Think back to the concept of implicit function in calculus. The idea here is that sometimes functions are not defined cleanly (explicitly) by a formula in which the dependent variable (on one side) is expressed in terms of the independent variable and some constants (on the other side), as in the solution $x=x(t)=5e^{3t}-7e^{2t}$ of Example 1.2.4. For instance, you may be given the relation $x^2+y^2=5$ , which can be written in the form $G(x,y)=0$ , where $G(x,y)=x^2+y^2-5$ . The graph of this relation is a circle of radius $\sqrt{5}$ centered at the origin, and this graph does not represent a function. (Why?) However, this relation does define two functions implicitly: $y_1(x)=\sqrt{5-x^2}$ and $y_2(x)=-\sqrt{5-x^2}$ , both having domains $[-\sqrt{5},\sqrt{5}]$ .

Definition 1.2.2

A relation $F(x,y)=0$ is said to be an implicit solution of a differential equation involving x, y, and derivatives of y with respect to x if $F(x,y)=0$ defines one or more explicit solutions of the differential equation.

More advanced courses in analysis discuss when a relation actually defines one or more implicit functions. This involves a result called the Implicit Function Theorem. For now, just remember that even if you can't untangle a relation to get an explicit formula for a function, you can use implicit differentiation to find derivatives of any differentiable functions that may be buried in the relation.

When trying to solve differential equations, often we can't find an explicit solution and must be content with a solution defined implicitly.

Example 1.2.5 Verifying an Implicit Solution

We want to show that any function y that satisfies the relation $G(x,y)=x^2+y^2-5=0$ is a solution of the differential equation $\frac{dy}{dx}=-\frac{x}{y}$ .

First, we differentiate the relation implicitly, treating y as $y(x)$ , an implicitly defined function of the independent variable x:

$(1)\frac{d}{dx}G(x,y)=\frac{d}{dx}(x^2+y^2-5)=\frac{d}{dx}(0)=0(2)2x+\stackrel{\text{Chain Rule}}{\overbrace{2y\frac{dy}{dx}}}-\frac{d}{dx}(5)=0(3)2x+2y\frac{dy}{dx}=0.$

Now, assuming that $y\ne 0$ , we solve Eq. (3) for $\frac{dy}{dx}$ , getting $\frac{dy}{dx}=\frac{-2x}{2y}=-\frac{x}{y}$ and proving that any function defined implicitly by the relation above is a solution of our differential equation.

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Ordinary differential equations

Brent J. Lewis , ... Andrew A. Prudil , in Advanced Mathematics for Engineering Students, 2022

Exact differential equations

A first-order differential equation of the form

(2.15)

is called exact if its left-hand side is the exact differential

(2.16) $du=\frac{\partial u}{\partial x}dx+\frac{\partial u}{\partial y}dy$

of some function $u(x,y)$ . Then the differential equation in Eq. (2.15) can be written as $du=0$ , and by integration it follows that

(2.17)

Comparing Eq. (2.15) and Eq. (2.16), Eq. (2.15) is exact if

(2.18) $(a)\frac{\partial u}{\partial x}=M\mathrm{and}(b)\frac{\partial u}{\partial y}=N.$

If M and N have continuous first partial derivatives, then

(2.19) $\frac{\partial M}{\partial y}=\frac{{\partial }^{2}u}{\partial y\partial x}\text{and}\frac{\partial N}{\partial x}=\frac{{\partial }^{2}u}{\partial x\partial y}.$

By continuity of the two second derivatives,

(2.20)

In fact, Eq. (2.20) is not only necessary but also sufficient for Eq. (2.15) to be an exact differential.

On integrating Eq. (2.18)(a) with respect to x,

(2.21)

To determine $k(y)$ , we derive $\partial u/\partial y$ from Eq. (2.21), use Eq. (2.18)(b) to get $dk/dy$ , and then integrate.

Example 2.1.7

Solve

(2.22) $(3x^{2}y)dx+(x^3+3y^2)dy=0.$

Solution.

(i) Test for exactness. We have $M=3x^{2}y$ and $N=x^3+3y^2$ . Thus, $\frac{\partial M}{\partial y}=3x^2=\frac{\partial N}{\partial x}$ . Therefore, Eq. (2.22) is exact.

(ii) Implicit solution . From Eq. (2.21) the implicit solution is $u=\int Mdx+k(y)=\int \left(3x^{2}y\right)dx+k(y)=x^{3}y+k(y)$ . To find $k(y)$ , use Eq. (2.18)(b) such that $\frac{\partial u}{\partial y}=x^3+\frac{dk}{dy}=N=x^3+3y^2$ . Thus, one obtains $\frac{dk}{dy}=3y^2$ with the solution $k=y^3+c^{\prime }$ . The final solution is

(2.23) $u(x,y)=(x^{3}y+y^3)=c.\text{[answer]}$

(iii) Check implicit solution $u(x,y)=c$ . Differentiating Eq. (2.23) with respect to x gives $(3x^{2}y+x^3{y}^{\prime }+3y^2{y}^{\prime })=0$ . This latter expression simplifies to $3x^{2}y+(x^3+3y^2)y^{\prime }=0$ , which, in turn, yields Eq. (2.22) since $y^{\prime }=dy/dx$ .

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First-order differential equations

Henry J. Ricardo , in A Modern Introduction to Differential Equations (Third Edition), 2021

Summary

An easy type of first-order ODE to solve is a separable equation, one that can be written in the form $\frac{dy}{dx}=f(x)g(y)$ , where f denotes a function of x alone and g denotes a function of y alone. "Separating the variables" leads to the equation $\int \frac{dy}{g(y)}=\int f(x)dx$ . It is possible that you cannot carry out one of the integrations in terms of elementary functions or you may wind up with an implicit solution. Furthermore, the process of separation of variables may introduce singular solutions.

Another important type of first-order ODE is a linear equation, one that can be written in the form $a_1(x)y^{\prime }+a_0(x)y=f(x)$ , where $a_1(x)$ , $a_0(x)$ , and $f(x)$ are functions of the independent variable x alone. The standard form of such an equation is $\frac{dy}{dx}+P(x)y=Q(x)$ . The equation is called homogeneous if $Q(x)\equiv 0$ and nonhomogeneous otherwise. Any homogeneous linear equation is separable.

After writing a first-order linear equation in the standard form $\frac{dy}{dx}+P(x)y=Q(x)$ , we can solve it by the method of variation of parameters or by introducing an integrating factor, $\mu (x)=e^{\int P(x)dx}$ .

A typical first-order differential equation can be written in the form $\frac{dy}{dx}=f(x,y)$ . Graphically, this tells us that at any point $(x,y)$ on a solution curve of the equation, the slope of the tangent line is given by the value of the function f at that point. We can outline the solution curves by using possible tangent line segments. Such a collection of tangent line segments is called a direction field or slope field of the equation. The set of points $(x,y)$ such that $f(x,y)=C$ , a constant, defines an isocline, a curve along which the slopes of the tangent lines are all the same (namely, C). In particular, the nullcline (or zero isocline) is a curve consisting of points at which the slopes of solution curves are zero. A differential equation in which the independent variable does not appear explicitly is called an autonomous equation. If the independent variable does appear, the equation is called nonautonomous. For an autonomous equation the slopes of the tangent line segments that make up the slope field depend only on the values of the dependent variable. Graphically, if we fix the value of the dependent variable, say x, by drawing a horizontal line $x=C$ for any constant C, we see that all the tangent line segments along this line have the same slope, no matter what the value of the independent variable, say t. Another way to look at this is to realize that we can generate infinitely many solutions by taking any one solution and translating (shifting) its graph left or right. Even when we can't solve an equation, an analysis of its slope field can be very instructive. However, such a graphical analysis may miss certain important features of the integral curves, such as vertical asymptotes.

An autonomous first-order equation can be analyzed qualitatively by using a phase line or phase portrait. For an autonomous equation the points x such that $\frac{dy}{dx}=f(x)=0$ are called critical points. We also use the terms equilibrium points, equilibrium solutions, and stationary points to describe these key values. There are three kinds of equilibrium points for an autonomous first-order equation: sinks, sources, and nodes. An equilibrium solution y is a sink (or asymptotically stable solution) if solutions with initial conditions "sufficiently close" to y approach y as the independent variable tends to infinity. On the other hand, if solutions "sufficiently close" to an equilibrium solution y are asymptotic to y as the independent variable tends to negative infinity, then we call y a source (or unstable equilibrium solution). An equilibrium solution that shows any other kind of behavior is called a node (or semistable equilibrium solution). The First Derivative Test is a simple (but not always conclusive) test to determine the nature of equilibrium points.

Suppose that we have an autonomous differential equation with a parameter α. A bifurcation point ${\alpha }_0$ is a value of the parameter that causes a change in the nature of the equation's equilibrium solutions as α passes through the value ${\alpha }_0$ . There are three main types of bifurcation for a first-order equation: (1) pitchfork bifurcation; (2) saddle-node bifurcation; and (3) transcritical bifurcation.

When we are trying to solve a differential equation, especially an IVP, it is important to understand whether the problem has a solution and whether any solution is unique. The Existence and Uniqueness Theorem provides simple sufficient conditions that guarantee that there is one and only one solution of an IVP. A standard proof of this result involves successive approximations, or Picard iterations.

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Advances in Plasma-Grown Hydrogenated Films

Wilfried G.J.H.M. Van Sark , in Thin Films and Nanostructures, 2002

1.4.3.2 Hybrid PIC/MC–Fluid Model

As a first attempt to modify the code to be able to run simulations on SiH₄–H₂ discharges, a hybrid PIC/MC–fluid code was developed [264, 265]. It turned out in the simulations of the silane–hydrogen discharge that the PIC/MC method is computationally too expensive to allow for extensive parameter scans. The hybrid code combines the PIC/MC method and the fluid method. The electrons in the discharge were handled by the fluid method, and the ions by the PIC/MC method. In this way a large gain in computational effort is achieved, whereas kinetic information of the ions is still obtained.

The fluid model is a description of the RF discharge in terms of averaged quantities [268, 269]. Balance equations for particle, momentum, and/or energy density are solved consistently with the Poisson equation for the electric field. Fluxes described by drift and diffusion terms may replace the momentum balance. In most cases, for the electrons both the particle density and the energy are incorporated, whereas for the ions only the densities are calculated. If the balance equation for the averaged electron energy is incorporated, the electron transport coefficients and the ionization, attachment, and excitation rates can be handled as functions of the electron temperature instead of the local electric field.

Especially for the electrons, the fluid model has the advantage of a lower computational effort than the PIC/MC method. Their low mass (high values of the transport coefficients) and consequent high velocities give rise to small time steps in the numerical simulation (v∆t   <  ∆x ) if a so-called explicit method is used. This restriction is easily eliminated within the fluid model by use of an implicit method. Also, the electron density is strongly coupled with the electric field, which results in numerical instabilities. This requires a simultaneous implicit solution of the Poisson equation for the electric field and the transport equation for the electron density. This solution can be deployed within the fluid model and gives a considerable reduction of computational effort as compared to a nonsimultaneous solution procedure [179]. Within the PIC method, only fully explicit methods can be applied.

The disadvantage of the fluid model is that no kinetic information is obtained. Also, transport (diffusion, mobility) and rate coefficients (ionization, attachment) are needed, which can only be obtained from experiments or from kinetic calculations in simpler settings (e.g. Townsend discharges). Experimental data on silane–hydrogen Townsend discharges are hardly available; therefore PIC/MC simulations have been performed.

The quasi-Townsend discharge is an infinitely extended plasma, in which the electrons move in a constant, spatially homogeneous electric field E. The electrons collide with neutrals in a background gas of infinite extent. The constant electric field replaces the solution of the Poisson equation and makes the PIC method applicable for the electrons within reasonable computational limits. During the particle simulation in the quasi-Townsend discharge, averaged values, over all electron particles i, of the energy ϵ _i , diffusion constant D_i , mobility μ_i , ionization rate k _ion,i , and attachment rate k _att, _i are calculated. This results in values for the transport and rate coefficients: the electron transport coefficients D_e and μ_e , the ionization and attachment rates k _ion and k _att and the averaged electron energy 〈ϵ〉   =   3k_BT_e /2. All parameters are obtained for several values of the electric field E.

Subsequently, because the electron temperature is known as a function of the electric field E, the temperature T_e can be used instead of E as a parameter for coefficients and rates, by elimination of E. Thus, the coefficients are available both as a function of E and of T_e. Both the local electric field [225, 269] and the electron temperature [239, 268] have been used as parameters in fluid modeling.

A Townsend PIC simulation was performed for several gas mixtures (SiH₄ : H₂  =   100:0, 50:50, 20:80, and 0:100) [264, 265]. It was found that the dependence of the electron temperature on the electric field strongly depends on the mixture ratio. As a result, the rate coefficient parametrized by the electric field, k   =   k(E), also strongly depends on the mixture ratio. In contrast, it was found that the rate as a function of the electron energy, k   =   k(T_e ), shows only minor changes (≤ 2%) for different mixing ratios. In Figure 25 the ionization (Fig. 25a) and attachment (Fig. 25b) rates are given. Here, two often used coefficients in RF fluid modeling are also shown for reference: an argon ionization rate derived from Townsend experiments (Fig. 25a), and a CF₄ attachment coefficient [179] (Fig. 25b).

Fig. 25. Simulation results for quasi-Townsend discharges: (a) ionization coefficients for SiH₄, H₂, and Ar; (b) attachment coefficients for SiH₄ and CF₄.

For the following reasons the choice of the temperature T_e as the experimental parameter is more appropriate [268, 270]: (a) the high-energy tail of the energy distribution function of the electrons determines ionization versus attachment; (b) both in pure silane and hydrogen and in mixtures, T_e is a good parameter for the distribution function; (c) T_e as a function of the electric field varies strongly with the mixture ratio SiH₄/H₂.

Methods that compensate for nonequilibrium effects in the situation of E-parametrized coefficients are very complicated, and are sometimes not firmly grounded. Because the electron temperature also gives reasonable results without correction methods, the rate and transport coefficients were implemented as a function of the electron energy, as obtained from the PIC calculations presented in Figure 25.

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Applications related to ordinary and partial differential equations

Martha L. Abell , James P. Braselton , in Mathematica by Example (Sixth Edition), 2022

6.1.3 Nonlinear equations

Mathematica can solve or help solve a variety of nonlinear first-order equations that are typically encountered in the introductory differential equations course.

Example 6.6

Solve each: (a) $(\cos x+2x{e}^y)dx+(\sin y+x^2{e}^y-1)dy=0$ ; (b) $(y^2+2xy)dx-x^{2}dy=0$ .

Solution

(a) Notice that $(\cos x+2x{e}^y)dx+(\sin y+x^2{e}^y-1)dy=0$ can be written as $dy/dx=-(\cos x+2x{e}^y)/(\sin x+x^2{e}^y-1)$ . The equation is an example of an exact equation. A theorem tells us that the equation

$M(x,y)dx+N(x,y)dy=0$

is exact if and only if $\partial M/\partial y=\partial N/\partial x$ .

$\mathit{m}\mathbf{=}\mathbf{\text{Cos}}\mathbf{[}\mathit{x}\mathbf{]}\mathbf{+}\mathbf{2}\mathit{x}\mathbf{\text{Exp}}\mathbf{[}\mathit{y}\mathbf{]}\mathbf{;}$

$\mathit{n}\mathbf{=}\mathbf{\text{Sin}}\mathbf{[}\mathit{y}\mathbf{]}\mathbf{+}{\mathit{x}}^{\mathbf{\wedge }}\mathbf{2}\mathbf{\text{Exp}}\mathbf{[}\mathit{y}\mathbf{]}\mathbf{-}\mathbf{1}\mathbf{;}$

$\mathit{D}\mathbf{[}\mathit{m}\mathbf{,}\mathit{y}\mathbf{]}$

$\mathit{D}\mathbf{[}\mathit{n}\mathbf{,}\mathit{x}\mathbf{]}$

$2e^{y}x$

$2e^{y}x$

We solve exact equations by integrating. Let $F(x,y)=C$ satisfy $(y\cos x+$ $2x{e}^y)dx+(\sin y+x^2{e}^y-1)dy=0$ . Then,

$F(x,y)=\int (\cos x+2x{e}^y)dx=\sin x+x^2{e}^y+g(y),$

where $g(y)$ is a function of y.

The function f, satisfying $df=f_x(x,y)dx+f_y)x,y)dy$ , is called the potential function.

$\mathbf{\text{f1}}\mathbf{=}\mathbf{\text{Integrate}}\mathbf{[}\mathit{m}\mathbf{,}\mathit{x}\mathbf{]}$

$e^y{x}^2+\text{Sin}[x]$

We next find that $g^{\prime }(y)=\sin y-1$ so $g(y)=-\cos y-y$ . Hence a general solution of the equation is

$\sin x+x^2{e}^y-\cos y-y=C.$

$\mathbf{\text{f2}}\mathbf{=}\mathit{D}\mathbf{[}\mathbf{\text{f1}}\mathbf{,}\mathit{y}\mathbf{]}$

$e^y{x}^2$

$\mathbf{\text{f3}}\mathbf{=}\mathbf{\text{Solve}}\mathbf{[}\mathbf{\text{f2}}\mathbf{+}\mathit{c}\mathbf{=}\mathbf{=}\mathit{n}\mathbf{,}\mathit{c}\mathbf{]}$

$\{\{c\to -1+\text{Sin}[y]\}\}$

$\mathbf{\text{Integrate}}\mathbf{[}\mathbf{\text{f3}}\mathbf{[}\mathbf{[}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{]}\mathbf{]}\mathbf{,}\mathit{y}\mathbf{]}$

$-y-\text{Cos}[y]$

We confirm this result with DSolve. Notice that Mathematica warns us that it cannot solve for y explicitly, and returns the same implicit solution obtained by us.

$\mathbf{\text{mf}}\mathbf{=}\mathit{m}\mathbf{\text{/.}}\mathit{y}\mathbf{\to }\mathit{y}\mathbf{[}\mathit{x}\mathbf{]}\mathbf{;}$

$\mathbf{\text{nf}}\mathbf{=}\mathit{n}\mathbf{\text{/.}}\mathit{y}\mathbf{\to }\mathit{y}\mathbf{[}\mathit{x}\mathbf{]}\mathbf{;}$

$\mathbf{\text{sol}}\mathbf{=}\mathbf{\text{DSolve}}\mathbf{[}\mathbf{\text{mf}}\mathbf{+}\mathbf{\text{nf}}{\mathit{y}}^{\mathbf{\prime }}\mathbf{[}\mathit{x}\mathbf{]}\mathbf{=}\mathbf{=}\mathbf{0}\mathbf{,}\mathit{y}\mathbf{[}\mathit{x}\mathbf{]}\mathbf{,}\mathit{x}\mathbf{]}$

$\text{Solve}[e^{y[x]}{x}^2-\text{Cos}[y[x]]+\text{Sin}[x]-y[x]==C[1],y[x]]$ .

Graphs of several solutions using the values of C generated in cvals are graphed with ContourPlot in Fig. 6.8.

Figure 6.8. (a) Graphs of several solutions of $(\cos x+2x{e}^y)dx+(\sin y+x^2{e}^y-1)dy=0$ . (b) Graphs of several solutions of $(y^2+2xy)dx-x^{2}dy=0$ (University of Arkansas colors).

$\mathbf{\text{sol2}}\mathbf{=}\mathbf{\text{sol}}\mathbf{[}\mathbf{[}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{]}\mathbf{]}\mathbf{\text{/.}}\mathit{y}\mathbf{[}\mathit{x}\mathbf{]}\mathbf{\to }\mathit{y}$

$e^y{x}^2-y-\text{Cos}[y]+\text{Sin}[x]$

$\mathbf{\text{cvals}}\mathbf{=}\mathbf{\text{Table}}\mathbf{[}\mathbf{\text{sol2}}\mathbf{\text{/.}}\mathbf{\{}\mathit{x}\mathbf{\to }\mathbf{-}\mathbf{3}\mathbf{\text{Pi}}\mathbf{/}\mathbf{2}\mathbf{,}\mathit{y}\mathbf{\to }\mathit{i}\mathbf{\}}\mathbf{,}\mathbf{\{}\mathit{i}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{6}\mathbf{\text{Pi}}\mathbf{,}\mathbf{6}\mathbf{\text{Pi}}\mathbf{/}\mathbf{24}\mathbf{\}}\mathbf{]}\mathbf{;}$

$\mathbf{\text{ContourPlot}}\mathbf{[}\mathbf{\text{sol2}}\mathbf{,}\mathbf{\{}\mathit{x}\mathbf{,}\mathbf{-}\mathbf{3}\mathbf{\text{Pi}}\mathbf{,}\mathbf{3}\mathbf{\text{Pi}}\mathbf{\}}\mathbf{,}\mathbf{\{}\mathit{y}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{6}\mathbf{\text{Pi}}\mathbf{\}}\mathbf{,}\mathbf{\text{Contours}}\mathbf{\to }\mathbf{\text{cvals}}\mathbf{,}$

$\mathbf{\text{ContourShading}}\mathbf{\to }\mathbf{\text{False}}\mathbf{,}\mathbf{\text{Axes}}\mathbf{\to }\mathbf{\text{Automatic}}\mathbf{,}\mathbf{\text{Frame}}\mathbf{\to }\mathbf{\text{False}}\mathbf{,}$

$\mathbf{\text{AxesOrigin}}\mathbf{\to }\mathbf{\{}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{\}}\mathbf{,}\mathbf{\text{ContourStyle}}\mathbf{\to }\mathbf{\text{CMYKColor}}\mathbf{[}\mathbf{.}\mathbf{07}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{.}\mathbf{68}\mathbf{,}\mathbf{.}\mathbf{32}\mathbf{]}\mathbf{]}$

(b) We can write $(y^2+2xy)dx-x^{2}dy=0$ as $dy/dx=(y^2+2xy)/x^2$ . A first-order equation is homogeneous if it can be written in the form $dy/dx=F(y/x)$ . Homogeneous equations are reduced to separable equations with either the substitution $y=ux$ or $x=vy$ . In this case, we have that $dy/dx={(y/x)}^2+2(y/x)$ , so the equation is homogeneous.

Let $y=ux$ . Then, $dy=udx+xdu$ . Substituting into $(y^2+2xy)dx-x^{2}dy=0$ and separating gives us

$$\begin{gathered} (y^2+2xy)dx-x^{2}dy=0(u^2{x}^2+2u{x}^2)dx-x^2(udx+xdu)=0(u^2+2u)dx-(udx+xdu)=0(u^2+u)dx=xdu \\ \frac{1}{u(u+1)}du=\frac{1}{x}dx. \end{gathered}$$

Integrating the left- and right-hand sides of this equation with Integrate,

$\mathbf{\text{Integrate}}\mathbf{[}\mathbf{1}\mathbf{/}\mathbf{(}\mathit{u}\mathbf{(}\mathit{u}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{)}\mathbf{,}\mathit{u}\mathbf{]}$

$\text{Log}[u]-\text{Log}[1+u]$

$\mathbf{\text{Integrate}}\mathbf{[}\mathbf{1}\mathbf{/}\mathit{x}\mathbf{,}\mathit{x}\mathbf{]}$

$\text{Log}[x]$

exponentiating, resubstituting $u=y/x$ , and solving for y gives us

$$\begin{gathered} \ln |u|-\ln |u+1|=\ln |x|+C \\ \frac{u}{u+1}=Cx \\ \frac{\frac{y}{x}}{\frac{y}{x}+1}=Cx \\ y=\frac{C{x}^2}{1-Cx}. \end{gathered}$$

$\mathbf{\text{sol1}}\mathbf{=}\mathbf{\text{Solve}}\mathbf{[}\mathbf{(}\mathit{y}\mathbf{/}\mathit{x}\mathbf{)}\mathbf{/}\mathbf{(}\mathit{y}\mathbf{/}\mathit{x}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{\text{==}}\mathit{c}\mathit{x}\mathbf{,}\mathit{y}\mathbf{]}$

$\{\{y\to -\frac{c{x}^2}{-1+cx}\}\}$

We confirm this result with DSolve, and then graph several solutions with Plot in Fig. 6.8 (b).

$\mathbf{\text{sol2}}\mathbf{=}\mathbf{\text{DSolve}}\mathbf{[}\mathit{y}{\mathbf{[}\mathit{x}\mathbf{]}}^{\mathbf{\wedge }}\mathbf{2}\mathbf{+}\mathbf{2}\mathit{x}\mathit{y}\mathbf{[}\mathit{x}\mathbf{]}\mathbf{-}{\mathit{x}}^{\mathbf{\wedge }}\mathbf{2}{\mathit{y}}^{\mathbf{\prime }}\mathbf{[}\mathit{x}\mathbf{]}\mathbf{=}\mathbf{=}\mathbf{0}\mathbf{,}\mathit{y}\mathbf{[}\mathit{x}\mathbf{]}\mathbf{,}\mathit{x}\mathbf{]}$

$\{\{y[x]\to -\frac{x^2}{x-C[1]}\}\}$

$\mathbf{\text{toplot}}\mathbf{=}\mathbf{\text{Table}}\mathbf{[}\mathbf{\text{sol2}}\mathbf{[}\mathbf{[}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{]}\mathbf{]}\mathbf{\text{/.}}\mathit{C}\mathbf{[}\mathbf{1}\mathbf{]}\mathbf{\to }\mathit{i}\mathbf{,}\mathbf{\{}\mathit{i}\mathbf{,}\mathbf{-}\mathbf{5}\mathbf{,}\mathbf{5}\mathbf{\}}\mathbf{]}\mathbf{;}$

$\mathbf{\text{Plot}}\mathbf{[}\mathbf{\text{Tooltip}}\mathbf{[}\mathbf{\text{toplot}}\mathbf{]}\mathbf{,}\mathbf{\{}\mathit{x}\mathbf{,}\mathbf{-}\mathbf{5}\mathbf{,}\mathbf{5}\mathbf{\}}\mathbf{,}\mathbf{\text{PlotRange}}\mathbf{\to }\mathbf{\{}\mathbf{-}\mathbf{5}\mathbf{,}\mathbf{5}\mathbf{\}}\mathbf{,}$

$\mathbf{\text{AspectRatio}}\mathbf{\to }\mathbf{\text{Automatic}}\mathbf{,}\mathbf{\text{PlotStyle}}\mathbf{\text{-}}\mathrm{>}\mathbf{\text{CMYKColor}}\mathbf{[}\mathbf{.}\mathbf{07}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{.}\mathbf{68}\mathbf{,}\mathbf{.}\mathbf{32}\mathbf{]}\mathbf{]}$   □

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Integrated Population Biology and Modeling, Part B

Narayan Behera , in Handbook of Statistics, 2019

2.2 Model Specification

There are n different values of an environmental variable corresponding to different patches in the environment of development and the environment of selection. A juvenile develops in patch x and obtains the genotypic value g(x) according to the wild-type reaction norm g. In generation t, the N _x (t) juveniles arrive in patch x. In generation t, the N _x (t) juveniles arrive in patch x and are subject to density-dependent number regulation. The probability of survival is u _x   =   exp[−  α _x N _x ] where α _x is the sensitivity to density regulation in patch x. A surviving individual migrates from its development patch x to a selection patch y with a conditional probability m _{y  |  x} where $\sum _{y=1}^n{m}_{y\mid x}=1$ for each x. The number of individuals in patch y after migration and before selection equals

(1) $N_{1,y}\left(t\right)=\sum _{x=1}^n{N}_x\cdot m_{y\mid x}.$

In patch y, Gaussian optimizing selection on the phenotypic trait g(x) occurs. The viability of an individual that developed in patch x (with phenotype g(x)) and moved to selection patch y is given by

(2) $w_{\mathit{xy}}^g\equiv W_y\left[g\left(x\right)\right]=exp\left\{-s{\left(\theta \left(y\right)-g\left(x\right)\right)}^2\right\}$

where θ(y) is the optimum trait value in patch y and s  =   1/(2σ ²) measures the intensity of optimizing selection in Gaussian selection of width σ ². After selection the number of individuals in selection patch y is given by

(3) $N_{2,y}\left(t\right)=N_{1,y}{w}_{\mathit{xy}}^g.$

Density-dependent number regulation after selection leads to the probability of survival as

(4) $z_y\equiv Z_y\left[N_{2,y}\left(t\right)\right]=exp\left\{-{\gamma }_y{N}_{2,y}\right\}$

where γ _y is the sensitivity in patch y to density regulation after selection. The number of individuals in patch y after selection equals

(5) $N_{3,y}=N_{2,y}{z}_y.$

Let d _{x  |  y} be the dispersal probability of a zygote from selection patch y to development patch x. Each surviving individual reproduces with fecundity F. The number of zygotes arriving in development patch x in generation t  +   1 equals

(6) $N_x\left(t+1\right)=\sum _{y=1}^n{N}_{3,y}{d}_{x\mid y}F.$

2.2.1 Fitness

In the zygote pool model, the population of zygotes is fully mixed before dispersing to the development habitats x. The dispersal probability to a patch x is equal for each patch y and equal to the frequency of occurrence of the patches x. The population is fully mixed in the zygote pool. So the recurrence equations of the numbers of each genotype can be written down from zygote pool to zygote pool.

At evolutionary equilibrium, the expectation of fitness W _g for a particular genotype g can be found from the recurrence equations (Sasaki and de Jong, 1999) as

(7) $E\left[W_g\right]={\sum }_x{\sum }_y{d}_x{u}_x^{\ast }{m}_{y\mid x}{v}_y^{\ast }{w}_{\mathit{xy}}^{g^{\ast }}{z}_y^{\ast }{d}_{x\mid y}F,$

where the symbol * refers to the individual number at evolutionary equilibrium. Weak selection at stable equilibrium number leads to a compromise ESS reaction norm defined by

(8) $g{\left(x\right)}^{\ast }={\sum }_y{m}_{y\mid x}{v}_y^{\ast }{w}_{\mathit{xy}}^{g^{\ast }}{z}_y^{\ast }{d}_{x\mid y}\theta \left(y\right)/{\sum }_y{m}_{y\mid x}{v}_y^{\ast }{w}_{\mathit{xy}}^{g^{\ast }}{z}_y^{\ast }{d}_{x\mid y}$

The evolved compromise reaction norm is weighted toward the optimum of the selection environment with least stringent density regulation, highest migration, and dispersal. Adults leaving their patch of development are counted. The total number of each genotype after a one-generation interval is tracked. Fitness is defined as the expected number of adult offspring of adult individuals leaving their patch of development for each genotype. We use the population growth rate of each population of one genotype. As these are haploid populations, this is an adequate measure of fitness.

The total number leaving any development patch x is N _x (t) in generation t; in the next generation t  +   1 the total number leaving any development patch k and descendant of an individual leaving patch x in generation t equals

(9) $N_x\left(t+1\right)=N_x\left(t\right){\sum }_k{\sum }_y\left(m_{y\mid x}{v}_y{w}_{x,y}^g{z}_{y}F\right)d_{k\mid y}.$

At stable equilibrium, the total number becomes

(10) $N^{\ast }={\sum }_x{N}_x^{\ast }\cdot {\sum }_k{\sum }_y\left(m_{y\mid x}{v}_y^{\ast }{w}_{x,y}^g{z}_y^{\ast }F{d}_{k\mid y}\right).$

The condition giving the ESS reaction norm is that no mutant can invade: the growth rate of any mutant is less than 1. Genotypic fitness is defined as the expected contribution to the next generation after averaging over all individuals of a genotype in the present generation. The number of individuals of the genotype g leaving any development patch x in generation t is denoted as N _x ^g (t). The expected contribution to the next generation of individuals of genotype g equals the growth rate of the population of genotype g.

The expected fitness W ^g   = N ^g (t  +   1)/N ^g (t). Averaging over all patch types, the fitness of the population becomes

(11) $W_g={\sum }_x{\sum }_y{\sum }_k\left(N_x^{\ast }{m}_{y\mid x}{v}_y^{\ast }{w}_{\mathit{xy}}^{g^{\ast }}{z}_y^{\ast }F{d}_{k\mid y}\right)/{\sum }_x{N}_x^{\ast }$

The condition for ESS reaction norm is that no rare mutant can invade in a population. The fitness of the resident genotype is at maximum and equal to one. All individuals have equal probability to undergo mutation at a certain very low rate. The population size is not affected by invasion, so also the density-dependent viabilities. The condition for the resident strategy of genotypic value g ^r to be a globally stable ESS is that for all mutant strategy g ^m , the mutant fitness is less than the fitness of one of the resident strategy g ^r . The ESS genotypic value involves a compromise between different optima.

The fitness is at a maximum at g(x )*. The implicit solution for the evolved reaction norm is given by

(12) $g{\left(x\right)}^{\ast }={\sum }_y{\sum }_k{m}_{y\mid x}{w}_{\mathit{xy}}^{g^{\ast }}{z}_y^{\ast }{d}_{k\mid y}\theta \left(y\right)/{\sum }_y{\sum }_k{m}_{y\mid x}{w}_{\mathit{xy}}^{g^{\ast }}{z}_y^{\ast }{d}_{k\mid y}$

With weak Gaussian selection, fitness has a maximum at g(x)*. g(x)* is an ESS under weak selection.

2.2.2 Density-Dependent Number Regulation

A "source" environment is a patch where food supply is high so that an individual survives with high probability. Conversely, food supply is low in a "sink" environment so that density regulation is large leading to lower survival probability of a population. The presence of "source" and "sink" environments highly influences the evolved reaction norm. In order to appreciate the effect of source and sink environments, it is useful to define the "flow" of individuals through the patches of environments of development and of selection. The flow summarizes the relative frequency of organisms meeting each sequence of patches y and x, according to

(13) $f\left(y|x\right)=m_{y\mid x}{v}_y^{\ast }{w}_{\mathit{xy}}^{g^{\ast }}{z}_y^{\ast }{d}_{x\mid y}/\left({\sum }_y{m}_{y\mid x}{v}_y^{\ast }{w}_{\mathit{xy}}^{g^{\ast }}{z}_y^{\ast }{d}_{x\mid y}\right)$

The flow is counted from just before migration of adults in generation t, to just before migration of adults in generation t  +   1. Flow acts as if it is the biological frequency of the environment combinations.

Given the definition of flow

(14) $g{\left(x\right)}^{\ast }={\sum }_{y}f\left(y|x\right){\theta }_y=E\left[{\theta }_y|x\right]$

in all models with equal selection intensity s in all environments y, the difference is in the definition of the flow. In the above E denotes the expected value.

Equal viability due to density-dependent number regulation before and after selection extinguishes the effect of density-dependence on selection. If z _y *   = z* and v _y *   = v _y in all the patches of the environment of selection, the flow becomes, for Gaussian selection:

(15) $f\left(y|x\right)=m_{y\mid x}{w}_{\mathit{xy}}^{g^{\ast }}{d}_{x\mid y}/\left({\sum }_y{m}_{y\mid x}{w}_{\mathit{xy}}^{g^{\ast }}{d}_{x\mid y}\right).$

The evolved phenotype g(x) is the same as for a pure genotypic frequency model, given we are using Gaussian optimizing selection.

2.2.3 Evolved Reaction Norm

We define the linear reaction norm as g(x)   = g ₀  + g ₁ x where g ₀ is the reaction norm height, g ₁ is the reaction norm slope, and x refers to the environment of development. The optimal reaction norm is defined as c(y)   = c ₀  + c ₁ y where c ₀ is the optimal reaction norm height, c ₁ is the reaction norm slope, and y denotes the environment of selection. Let r(x,y) denote the migration correlation between the patch of development x and the patch of selection y. If selection is predictable, r(x,y)   =   1, then the evolved reaction norm is given by g ₀  = c ₀ and g ₁  = c ₁. But, as the selection becomes unpredictable, 0   ≤ r(x,y)   ≤   1, the evolved reaction norm might differ from the optimal reaction norm. Weak selection leads to the bet-hedging of the reaction norm, while strong selection gives polymorphism. The slope ${\overline{g}}_1$ of the evolved mean reaction norm $\overline{g}={\overline{g}}_0+{\overline{g}}_1$ x differs, depending upon the unpredictability of selection and quadratic or Gaussian optimizing fitness function. When genotypic reaction norms are linear and optimum reaction norm is linear (meaning only the first two coefficients exist in the genetic variation), the evolved mean genotypic reaction norm coefficients are given by (de Jong, 1999)

(16a) $g_0=c_0+c_1\overline{y}-{\overline{g}}_1\overline{x}$

But now the density dependence influences the distribution of individuals over the patches. The optimal reaction norm is always obtained when the food supply is uniform over the patches and migration is symmetric. The deviation from the optimal reaction norm occurs due to the asymmetry in the effective migration distribution. This is caused by two factors. One is the mirroring of migration due to the edges of the environment. This happens when the number of patches is finite. It can be better illustrated by a simple example. Suppose the migration width is three, and there is equal a priori probability of migration. If the straight migration occurs at the right boundary, there is no possibility of migration to one patch right of the boundary. This migration probability will be mirrored to the left patch of the boundary so that the probability of migration to this patch will be 0.666. This can cause asymmetry in the migration distribution. The other factor is the density-dependent number regulation due to nonuniform food supply. The nonuniform food supply will cause differential density regulation over the patches. This can create asymmetry in the migration distribution.

The predicted ESS g ₀ and g ₁ values depend on the number of individuals at evolutionary equilibrium. But this number cannot be found analytically. Furthermore, the small deviations of the evolved g ₀ and g ₁ values from the optimal c ₀ and c ₁ values are difficult to find as selection is weak and the model is individual-based. So it seems best to develop a mutation model where deviations in g ₀ and g ₁ are introduced with low probability.

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Finding Implicit and Explicit Solutions to Initial Value Problems

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