On Hybrid-Function Solutions of the Lotka–Volterra Equations

Abstract

The classical Lotka–Volterra predator–prey system is often used in modeling species competition. The two-species nonlinear system is expressed in terms of a single positive coupling parameter $\lambda$. Based on a standard functional transformation, a novel $\lambda$-invariant Hamiltonian yields a system of two partially uncoupled first-order hybrid-function ODEs, albeit with one being linear. An exact single quadrature solution that is valid for any value of $\lambda$ and the system’s energy is derived. In the particular case of $\lambda =1$, the ODE system completely uncouples. One ODE is autonomous. An exact analytic quadrature solution is derived that is predicated on the exact turning-point solutions. It is expressed in terms of the Lambert W function and must be evaluated numerically. Exact time-dependent solutions are presented for each individual species separately. In the case of $\lambda \ne 1$ an accurate practical approximation uncoupling the nonlinear system is proposed and solutions are provided in terms of explicit quadratures together with high-energy asymptotic solutions. An exact analytic expression for the system’s oscillation period that is valid for any value of $\lambda$ and orbital energy is derived in terms of a dimensionless energy function.

1. Introduction

The historic Lotka–Volterra (“LV”) predator–prey system of two coupled first-order nonlinear differential equations was first investigated in ecological and chemical systems [1,2]. This idealized model describes the competition of two isolated coexisting species: a ‘prey’ population evolves while feeding on an unrestricted resource supply, whereas ‘predators’ interact by exclusively feeding on prey, either through direct predation or as parasites. As a result the respective populations exhibit undamped periodic oscillations as a function of time with a period that depends on the species interaction rates.

This idealized two-species model has further been generalized to interactions between multiple coexisting species in biological mathematics [3], ecology [4], virus propagation [5], and also in molecular vibration–vibration energy transfers [6].

2. Normalized Equations and Single Coupling Parameter

The classical nonlinear LV model is based on four time-independent, positive, and constant rates with two representing species self-interaction, i.e., natural exponential growth rate $\alpha$ and decay rate $\delta$ per individual of the respective prey and predator populations, and two other rates characterizing inter-species interaction.

Without any loss of generality, the standard LV system of two coupled 1st-order ordinary differential equations (ODEs) can be simplified by simultaneously scaling the predator and prey populations together with time $t^{\prime }$ through a dimensionless time factor $t^{\prime }/\sqrt{\alpha \delta }\to t$. The system is shown to only depend on a positive coupling parameter $\lambda$, ratio of the respective growth and decay rates of each species taken separately [7], defined as

$$\lambda =\sqrt{{\displaystyle \frac{\alpha }{\delta }}}.$$

(1)

The respective instantaneous prey and predator populations, labeled u and v, both ⩾0, are assumed to be continuous functions of time ($u,v\in \mathbb{R}$). A normalized form of the LV system is obtained as a set of two coupled 1st-order nonlinear ODEs solely depending on this single coupling ratio $\lambda$. The time evolution of the two species is modeled as a system of two coupled autonomous nonlinear ODEs where the “dot” on $\dot{u}$ and $\dot{v}$ indicates a derivative with respect to the dimensionless time t

$$\begin{array}{cc}\hfill \dot{u}& =\lambda u\left(1-v\right),\mathrm{for}\mathrm{prey}\hfill \end{array}$$

(2a)

$$\begin{array}{cc}\hfill \dot{v}& ={\displaystyle \frac{1}{\lambda }}v\left(u-1\right).\mathrm{for}\mathrm{predators}\hfill \end{array}$$

(2b)

Numerous solutions of system (2) have been developed, including trigonometric series [8], mathematical transformations [9], Taylor series expansions [10], perturbation techniques [11], Lambert W-functions [12], and numeric-analytic techniques [13].

In the sudden absence of coupling between species, the prey population grows at an exponential rate $\lambda$, while predators similarly decay at an inverse rate $-1/\lambda$ from their respective positive initial values. Remarkably, the normalized ODE system (2) is invariant in the transformation $u\to v$ together with $\lambda \to -1/\lambda$. This fundamental property, subsequently referred to as “$\lambda$ invariance”, is extensively used throughout to considerably simplify the LV problem analysis.

Since the original publications [1,2], the non-trivial system (2) has been known to possess a dynamical invariant or “constant of motion K”, expressed here in $\lambda$-invariant form:

$$\frac{1}{\lambda }u+\lambda v-\ln \left(u^{\frac{1}{\lambda }}{v}^{\lambda }\right)=K$$

(3)

The objective of this research is to formulate the LV problem in a simple form while attempting to uncouple the basic ODE system: the introduction of new “hybrid species” enables uncoupling the LV system into 2 ODEs with one being linear.

In the following sections, a particular functional transformation combined with a suitable linear change of variables defines a novel λ-invariant Hamiltonian based on these hybrid variables, thereby reducing the system (2) to a new set of two 1st-order ODEs with one being linear. As a result, an exact analytic solution is derived for one hybrid function in terms of a simple quadrature. An original approximate method uncouples the system, yielding approximate analytic quadrature solutions of the LV problem. The LV oscillation period as a function of the coupling parameter $\lambda$ and the system’s energy h is further presented. At high energies, closed-form asymptotic solutions are derived.

In the special case of $\lambda =1$, the LV system completely uncouples into two 1st-order ODEs, with one being linear and the other being autonomous. An exact analytic solution is derived for each hybrid function separately in terms of a single quadrature expressed in terms of the Lambert W function. An exact analytic solution of the LV system for each individual prey and predator species $u\left(t\right)$ and $v\left(t\right)$ is also derived. The exact population oscillation period is further presented in terms of a dimensionless energy function.

The $\lambda \ne 1$ case represents a physically more realistic case in ecology, whereas the $\lambda =1$ case can also be interpreted as a (rare) competitive case in which the prey is another predator: it corresponds to the ecological condition $\alpha =\delta$, i.e., equal intrinsic growth and decay rates of prey and predator. While non-generic, it is more than a mathematical idealization since it is a special ecologic condition that provides the one case where the uncoupling (23) is exact, serving as a benchmark for the accuracy of the general case $\lambda \ne 1$ approximation. Lastly, it should be mentioned that an exact solution of the LV system has previously been derived [14] in the case when $\lambda =-1$: this non-ecologic condition precludes population oscillation.

3. Solutions with Hybrid Predator–Prey Species

A logarithmic functional transformation originally introduced by Kerner [15] proposes to modify the coupling between the respective species through a change of variables according to

$$y=ln\left(u\right)\mathrm{and}x=ln\left(v\right)\mathrm{with}y\in (-\infty ,+\infty ),x\in (-\infty ,+\infty )$$

(4)

The LV system (2) for the respective “logarithmic” prey-like and predator-like species $y\left(t\right)$ and $x\left(t\right)$ becomes

$$\begin{array}{c}\dot{y}=\lambda (1-e^x)\hfill \\ \dot{x}={\displaystyle \frac{1}{\lambda }}(e^y-1)\hfill \end{array}$$

(5)

Similar to (2) this $\lambda$-invariant system (5) admits a primary conservation integral H expressed as the linear combination of two positive convex functions

$$H(x,y)=\lambda (e^x-x-1)+{\displaystyle \frac{1}{\lambda }}(e^y-y-1)$$

(6)

As already established [16,17], $H(x,y)$ is the time-independent Hamiltonian of the LV system since (5) satisfies Hamilton’s equations with x as the coordinate conjugate to the canonical momentum y. Equation (6) expresses the conservative coupling between species $x\left(t\right)$ and $y\left(t\right)$. It is further rendered $\lambda$-invariant by introducing a scaled Hamiltonian $h(x,y)$ with total constant positive energy simply labeled h according to

$$H\left(x,y\right)=\left(\lambda +{\displaystyle \frac{1}{\lambda }}\right)h\left(x,y\right)$$

(7)

A $\lambda$-invariant linear 1st-order ODE between the species $x\left(t\right)$ and $y\left(t\right)$ is introduced by further combining the system (5) with (6) and (7)

$$\dot{x}-\dot{y}-\left(\lambda x+{\displaystyle \frac{y}{\lambda }}\right)=\left(\lambda +{\displaystyle \frac{1}{\lambda }}\right)h$$

(8)

This linear differential Equation (8) suggests introducing a final λ-invariant linear transformation of the set {x(t), y(t)} to a new set {ξ(t), η(t)} representing a symbiotic association between predator and prey species in which each hybrid function of the new set of canonical Hamiltonian variables {ξ(t), η(t)} is a linear combination of the original predator and prey populations. This is the unique combination of x and y that converts the linear ODE (8) into the canonical form $\dot{\eta }=\xi +h$ (see (17a)) by requiring the coefficient structure of (8) to match a momentum or coordinate pair of Hamilton equations,

$$\begin{array}{cc}\hfill \xi & ={\displaystyle \frac{\lambda x+\frac{1}{\lambda }y}{\lambda +\frac{1}{\lambda }}}\hfill \end{array}$$

(9a)

$$\begin{array}{cc}\hfill \eta & ={\displaystyle \frac{x-y}{\lambda +\frac{1}{\lambda }}}\hfill \end{array}$$

(9b)

The original Hamiltonian (6) together with Equation (9) and the linear transformation (7) then becomes

$$h(\eta ,\xi )={\displaystyle \frac{\lambda e^{\frac{\eta }{\lambda }}+\frac{1}{\lambda }{e}^{-\lambda \eta }}{\lambda +\frac{1}{\lambda }}}{e}^{\xi }-\xi -1$$

(10)

Here $h\left(\eta ,\xi \right)$ is a new Hamiltonian for the coordinate $\eta$ and conjugate momentum $\xi$. Notice that, for small amplitudes, $h\left(\eta ,\xi \right)$ is a harmonic oscillator Hamiltonian. Upon further introducing the following $\lambda$-invariant G function

$$G_{\lambda }\left(\eta \right)={\displaystyle \frac{\lambda e^{\frac{\eta }{\lambda }}+\frac{1}{\lambda }{e}^{-\lambda \eta }}{\lambda +\frac{1}{\lambda }}}\mathrm{with}{G}_{\lambda }\left(\eta \right)=G_{1/\lambda }(-\eta )(\lambda -\mathit{invariance})$$

(11)

the conservation relationship (10) between the conjugate functions η(t) and ξ(t) is recast into a compact form that naturally separates the variables,

$$G_{\lambda }\left(\eta \right)=(\xi +h+1)e^{-\xi }$$

(12)

In the following, a useful compact auxiliary function U(ξ) $⩾1$ appearing throughout is defined as

$$U\left(\xi \right)=(\xi +h+1)e^{-\xi }.$$

(13)

Even though still nonlinear, the conservation relationship (12) partially uncouples the ξ(t)-function from the η(t)-function, resulting in three essential G-function properties:

• The system’s energy $h⩾0$ is explicitly associated with the function $U\left(\xi \right)$ only;
• The positive function $G_{\lambda }\left(\eta \right)$ is a convex generalized hyperbolic cosine function reaching its minimum $G_{\lambda }=1$ at $\eta =0$ for any value of $\lambda$. Its inverse function $G_{\lambda }^{-1}$ exists, and, for any value of $\lambda$, (12) admits two respective positive and negative roots ${\eta }^{\pm }(\xi ,\lambda )$, functions of $\xi$ only, satisfying (The existence and uniqueness of $G_{\lambda }^{-1}$ follow directly from the convexity of $G_{\lambda }\left(\eta \right)$, which is a monotonic continuous function on each half line $(\infty ,0]$ and $[0,\infty )$ whose 2nd derivative ${G″}_{\lambda }\left(\eta \right)$ is strictly positive. Hence $G_{\lambda }\left(\eta \right)$ is strictly convex on all $\mathbb{R}$ and is a strict bijection onto $[1,\infty )$; hence, $G_{\lambda }^{-1}$ is its unique continuous inverse on this line.)

  $${\eta }^{\pm }(\xi ,\lambda )=G_{\lambda }^{-1}\left(U\left(\xi \right)\right)$$

  (14)
• Since the $\eta$ function is associated with the coupling ratio $\lambda$ only, $\lambda$ invariance of the G function (11) implies that, for a given $\lambda$, any positive solution ${\eta }^{+}(\xi ,\lambda )$ is directly derived from the negative solution associated with the ratio $1/\lambda$. Reciprocally,

  $${\eta }^{\pm }(\xi ,\lambda )=-{\eta }^{\mp }(\xi ,1/\lambda )$$

  (15)

From (12), the $\xi \left(t\right)$ function thus oscillates between the respective negative and positive roots ${\xi }^{-}\left(h\right)$ and ${\xi }^{+}\left(h\right)$, turning points of the equation $G_{\lambda }\left(0\right)=1$; i.e., $U\left(\xi \right)=1$. These are naturally expressed in exact closed form in terms of the Lambert W function [18] as

$${\xi }^{\pm }\left(h\right)=-(h+1)-W_k(-e^{-(h+1)}),$$

(16)

where the index k refers to the respective $k=0$ and $k=-1$ branches of the Lambert W function since $h\in \mathbb{R}$. The roots are displayed in Table 2 in Section 4 for several increasing values of h.

In the $\xi -\eta$ plane, (12) represents a symmetric closed-orbit mapping around the fixed point $(0,0)$ in closed form. On the $\eta =0$ horizontal axis this orbit is bounded by the limits ${\xi }^{-}\left(h\right)$ and ${\xi }^{+}\left(h\right)$, and, since U(ξ) admits a maximum $e^h$ located at $\xi =-h$, it is also bounded by the two respective positive and negative root solutions of the equation ${\eta }^{\pm }(-h,\lambda )=G_{\lambda }^{-1}\left(e^h\right)$. For any given energy h this orbit consists of two respective branches, ${\eta }^{+}(\xi ,\lambda )$ and ${\eta }^{-}(\xi ,\lambda )$, associated with the decay and growth of $\xi \left(t\right)$, the switch occurring at the tuning points, as displayed in Figure 1, where the respective values chosen are $h=2$ and coupling ratios $\lambda =2$ and $\lambda =1/2$. Per (12), the respective branches associated with the $\lambda$ and $1/\lambda$ mappings are readily observed to be mirror images of each other with respect to the $\eta =0$ axis.

Except when $\lambda =1$, algebraic solutions of (12) may generally not be obtained directly. However, for any value $\xi \in \{{\xi }^{-}\left(h\right),{\xi }^{+}\left(h\right)\}$, the two roots ${\eta }^{\pm }(\xi ,\lambda )$ of (12) may numerically be obtained through a standard Newton–Raphson algorithm: each root admits lower and upper bounds for any value of U(ξ), thereby ensuring algorithm convergence [7].

Lastly, upon inserting the linear transformation (9) into the modified LV system (5), or equivalently using the standard Hamilton equations with (10), a new semilinear system of coupled 1st-order ODEs is obtained:

$$\begin{array}{cc}\hfill \dot{\eta }& =\xi +h\hfill \end{array}$$

(17a)

$$\begin{array}{cc}\hfill \dot{\xi }& =-G_{\lambda }^{\prime }\left(\eta \right)e^{\xi }\hfill \end{array}$$

(17b)

The solution of the system (17), in which $G_{\lambda }^{\prime }\left(\eta \right)$ is the derivative $G_{\lambda }^{\prime }\left(\eta \right)=d{G}_{\lambda }/d\eta$, represents the time evolution of the hybrid functions $\eta \left(t\right)$ and $\xi \left(t\right)$. However, due to the linear transformation (9), the first coupled equation (17a) becomes linear since it directly expresses ODE (8). Remarkably, up to the constant energy h, the time derivative of the function $\eta \left(t\right)$ is directly equal to the instantaneous value of $\xi \left(t\right)$, thereby simplifying the solution of (17). The exact solution of the LV problem is then derived by integration of the linear ODE (17a) as a simple quadrature for $t\left(\xi \right)$, with time as a function of $\xi$. Upon using the initial conditions $\eta \left(0\right)=0$ and $\xi \left(0\right)={\xi }^{\pm }\left(h\right)$ when $t=0$, the exact LV solution corresponding to the respective negative and positive branches ${\eta }^{-}(\xi ,\lambda )$ and ${\eta }^{+}(\xi ,\lambda )$ becomes

$$t\left(\xi \right)={\int }_{{\xi }^{\pm }}^{\xi }{\displaystyle \frac{d{\eta }^{\pm }(x,\lambda )}{h+x}}$$

(18)

This quadrature does not diverge at $x=-h$ since the numerator of the differential $d\eta$ contains $U^{\prime }\left(x\right)=-(h+x)e^{-x}$. Upon using the same initial conditions for $\eta \left(0\right)$ and $\xi \left(0\right)$, (18) is expressed in terms of the continuous function ${\eta }^{\pm }(\xi ,\lambda )$ itself through a standard integration by parts in which the singularity at $\xi =-h$ is further eliminated by adding and subtracting the expression ${\displaystyle \frac{{\eta }^{\pm }(-h,\lambda )}{h+\xi }}$ in the integral. The final exact regular solution of the LV problem for any value of the coupling ratio $\lambda$ and any value of the orbital energy h is thus expressed as a simple quadrature over each of the two branches ${\eta }^{\pm }(\xi ,\lambda )$ that are solutions of (14) (Consider $G_{\lambda }\left(\eta \right)=U\left(\xi \right)$; since $U^{\prime }\left(\xi \right)$ vanishes at $\xi =-h$, taking the derivatives of both sides implies ${\eta }^{\prime }(-h)=0$ (obvious from the graph of Figure 1). Upon calling $f^{\pm }\left(x\right)$ the numerator of the integral in (19) so that $f^{\pm }(-h)=0$ and $f^{\prime \pm }(-h)=0$, then $f^{\pm }\left(x\right)={\displaystyle \frac{1}{2}}\eta {"}^{\pm }(-h,\lambda ){(x+h)}^2+O\left({(x+h)}^3\right)$. In the limit $\xi \to -h$, by l’Hopital rule, the 1st term in (19) vanishes since it is the definition of the derivative ${\eta }^{\prime \pm }(-h,\lambda )$, while the 2nd term is manifestly finite since $-h$ never coincides with the turning points. As $x\to -h$ the integrand of the (19) integral equals ${\displaystyle \frac{1}{2}}\eta {"}^{\pm }(-h,\lambda )$, which is finite. Therefore (19) is convergent at $x=-h$.)

$$t\left(\xi \right)={\displaystyle \frac{{\eta }^{\pm }(\xi ,\lambda )-{\eta }^{\pm }(-h,\lambda )}{h+\xi }}+{\displaystyle \frac{{\eta }^{\pm }(-h,\lambda )}{h+{\xi }^{\pm }}}+{\int }_{{\xi }^{\pm }}^{\xi }{\displaystyle \frac{{\eta }^{\pm }(x,\lambda )-{\eta }^{\pm }(-h,\lambda )}{{(h+x)}^2}}dx$$

(19)

This solution for $\lambda \ne 1$ is further analyzed below. Even though the oscillation of $\xi \left(t\right)$ is not explicitly expressed as a function of time t, the function t(ξ) being monotonic and continuous on each respective integration interval defined above, its inverse function $t^{-1}\left(\xi \right):\mathbb{R}\to \mathbb{R}$, defined by $t^{-1}\left(\xi \right)\to \xi \left(t\right)$, exists and is unique, monotonic, and continuous on each interval.

Numerical solutions for $\xi \left(t\right)$ and $\eta \left(t\right)$ are also obtained by integrating (17) using a standard fourth-order Runge–Kutta (RK4) method as presented in Figure 2 for values of h and $\lambda$ that are identical to those of Figure 1, together with initial conditions $\eta \left(0\right)$ and $\xi \left(0\right)$, defined above. The function $\xi \left(t\right)$ is observed to principally depend on two time constants: a quasi-exponential increase at a rate of order $\lambda$ followed by an exponential decrease at a rate $-1/\lambda$. Its amplitude ${\xi }^{+}\left(h\right)-{\xi }^{-}\left(h\right)$ only depends on the system’s energy. As expected from $\lambda$ invariance, the two functions $\xi \left(t\right)$ respectively corresponding to the coupling ratio $\lambda =2$ and its inverse $\lambda =1/2$ are mirrors of each other; so are the functions $\eta \left(t\right)$ but with the change $\eta \to -\eta$.

It may generally not be possible to algebraically solve (12) for $\eta (\xi ,\lambda )$ for insertion into the integral solution (19). A strategy consists of eliminating the $\eta$ dependence in (17b) and seeking an ODE for $\xi \left(t\right)$ only. An approximate relationship explicitly relating $G_{\lambda }\left(\eta \right)$ to its derivative ${G^{\prime }}_{\lambda }\left(\eta \right)$ and expressing the latter as an analytic function of $\xi$ only through (12) is derived below.

3.1. Case $\lambda =1$

This particular case is exactly solved since an explicit relationship exists between $G_1\left(\eta \right)$ and ${G^{\prime }}_1\left(\eta \right)$, namely (omitting the index for simplicity)

$$G^{\prime }\left(\eta \right)=\pm {(G^2-1)}^{1/2}$$

(20)

The G function (11) reduces to the hyperbolic cosine function; the conservation Equation (12) becomes a closed-form expression,

$$G\left(\eta \right)=\cosh \left(\eta \right)=(h+1+\xi )e^{-\xi }$$

(21)

The resulting $\xi$ – $\eta$ closed-orbit mapping is symmetric. On the $\xi$ axis, for any value of the energy h, the closed-form mapping is bounded by ${\xi }^{-}\left(h\right)$ and ${\xi }^{+}\left(h\right)$. The two symmetric branches ${\eta }^{\pm }\left(\xi \right)$ around the fixed point $(0,0)$ are explicitly expressed as

$${\eta }^{\mp }\left(\xi \right)=\pm {\cosh }^{-1}\left((h+1+\xi )e^{-\xi }\right)$$

(22)

The negative branch ${\eta }^{-}\left(\xi \right)$ is associated with the growth phase of $\xi \left(t\right)$, whereas ${\eta }^{+}\left(\xi \right)$ is associated with the $\xi \left(t\right)$ decay. The orbit is bounded by the turning points ${\xi }^{-}\left(h\right)$ and ${\xi }^{+}\left(h\right)$ on the $\eta =0$ horizontal axis, and, since $\eta \left(\xi \right)$ admits an extremum $\pm e^h$ located at $\xi =-h$, it is also bounded on the $\xi =0$ vertical axis by the two respective roots of the equation ${\eta }^{\mp }(-h)=\pm {\cosh }^{-1}\left(e^h\right)$.

Lastly, inserting (22) into (17) yields a new $\mathit{uncoupled}$ system of two 1st-order ODEs for each hybrid function taken separately, with the evolution of $\xi \left(t\right)$ represented by a 1st-order nonlinear autonomous ODE,

$$\begin{array}{cc}\hfill \dot{\eta }& =\xi +h,\hfill \end{array}$$

(23a)

$$\begin{array}{cc}\hfill \dot{\xi }& =\pm {\left({(\xi +h+1)}^2-e^{2\xi }\right)}^{1/2}.\hfill \end{array}$$

(23b)

The solution of system (23) represents the periodic time evolution of both functions $\eta \left(t\right)$ and $\xi \left(t\right)$. Remarkably, in this $\lambda =1$ case, the LV problem solution is simplified since a solution of the autonomous Equation (23b) only is required.

The linear Equation (23a) is directly solved by inserting $\eta \left(\xi \right)$ from (22) into the solution (19). Together with $U\left(\xi \right)$ defined in (13), the exact analytic solution on the interval ${\xi }^{-}⩽\xi ⩽{\xi }^{+}$ is thus expressed as a simple quadrature in terms of elementary functions

$$t\left(\xi \right)={\displaystyle \frac{{\cosh }^{-1}\left(e^h\right)-{\cosh }^{-1}\left(U\left(\xi \right)\right)}{h+\xi }}-{\displaystyle \frac{{\cosh }^{-1}\left(e^h\right)}{h+{\xi }^{-}}}+{\int }_{{\xi }^{-}}^{\xi }{\displaystyle \frac{{\cosh }^{-1}\left(e^h\right)-{\cosh }^{-1}\left(U\left(x\right)\right)}{{(h+x)}^2}}dx$$

(24)

By applying l’Hôpital’s rule, it is readily verified that the integrand in (24) is regular at $\xi =-h$. For a given energy h, solutions for $\xi \left(t\right)$ can be obtained by numerical integration of (24). The complete solution of the LV problem for $\lambda =1$ is finalized for $\eta \left(t\right)$ by inserting $\xi \left(t\right)$, derived above, into (22).

A numerical solution for $\xi \left(t\right)$ can also be obtained by integrating (23b) using a standard fourth-order Runge–Kutta (RK4) method. Figure 3 presents the $\xi \left(t\right)$ solution obtained by numerical integration for an energy $h=2$ with initial condition $\xi \left(0\right)={\xi }^{-}\left(h\right)$. The growth and decay phases of the even function $\xi \left(t\right)$ are symmetric relative to the half period $T\left(h\right)/2=t^{\ast }$ when $\xi \left(t^{\ast }\right)={\xi }^{+}\left(h\right)$ (see Section 5). The LV system solution is finalized for the two branches ${\eta }^{\mp }\left(t\right)$ by inserting $\xi \left(t\right)$, derived above, into (22) since $\eta \left(0\right)=0$.

Over the respective intervals ${\xi }^{-}⩽\xi \left(t\right)⩽{\xi }^{+}$ and ${\xi }^{+}⩾\xi \left(t\right)⩾{\xi }^{-}$ corresponding to the growth and decay phases of $\xi \left(t\right)$, an integral expression for $t\left(\xi \right)$ is readily obtained by performing the integration with the respective positive root (growth phase) and negative root (decay phase) in (23b), yielding the following quadrature solution:

$$\begin{array}{c}\hfill t\left(\xi \right)={\int }_{{\xi }^{-}}^{\xi }{\displaystyle \frac{dx}{\sqrt{{(x+h+1)}^2-e^{2x}}}},\end{array}$$

(25a)

$$\begin{array}{c}\hfill t\left(\xi \right)=2t^{\ast }-{\int }_{{\xi }^{-}}^{\xi }{\displaystyle \frac{dx}{\sqrt{{(x+h+1)}^2-e^{2x}}}}.\end{array}$$

(25b)

At the respective turning points ${\xi }^{-}\left(h\right)$ and ${\xi }^{+}\left(h\right)$, the integrand of (25) has a square root-type singularity, which is a stiff challenge in numerical integration. Yet, it is strictly continuous over the interval and the improper integral is convergent.

Together with (22), the exact integral solution (25) for the symmetric function $\xi \left(t\right)$ over the respective growth and decay intervals constitutes the final solution of the LV problem for the hybrid functions in this special $\lambda =1$ case.

The solution (25) is similar in form to an analytic quadrature solution derived by Evans and Findley (Equation (17) in [9]); however, the authors do not indicate how to solve their function’s numerical integration since it contains a square root singularity. The integral (25) lends itself to a more natural quadrature solution predicated on (16) and derived in Appendix A in terms of the bounded Lambert W function, which analytically removes the square root singularity from the denominator. Both integral formulations are equivalent, but the Lambert W substitution provides a justified numerical evaluation scheme.

3.2. Solutions for the Prey and Predator Species Populations in the Case of $\lambda =1$

Exact solutions for the time evolution of the prey and predator populations are derived by inserting the respective functions $\xi \left(t\right)$ and $\eta \left(t\right)$ obtained from (25) and (22) into the original definitions (9). This results in two uncoupled solutions for the original individual prey and predator species populations $u\left(t\right)$ and $v\left(t\right)$. Over the growth and decay phases of the $\xi \left(t\right)$ function, these exact uncoupled analytical solutions are expressed as

Interval $0⩽t⩽t^{\ast }$ ↦ interval ${\xi }^{-}⩽\xi \left(t\right)⩽{\xi }^{+}$, i.e., $\xi \left(t\right)$ growth phase

$$\begin{array}{ccc}\hfill u\left(t\right)& =\xi \left(t\right)+h+1+\sqrt{{(\xi \left(t\right)+h+1)}^2-e^{2\xi \left(t\right)}},\hfill & \hfill \mathrm{for}\mathrm{prey}\end{array}$$

(26a)

$$\begin{array}{ccc}\hfill v\left(t\right)& =\xi \left(t\right)+h+1-\sqrt{{(\xi \left(t\right)+h+1)}^2-e^{2\xi \left(t\right)}}.\hfill & \hfill \mathrm{for}\mathrm{predators}\end{array}$$

(26b)

with $t^{-1}\left(\xi \right)\to \xi \left(t\right)$ derived from (25a) together with $\xi \left(0\right)={\xi }^{-}\left(h\right)$.

Interval $t^{\ast }⩽t⩽2t^{\ast }$ ↦ interval ${\xi }^{+}⩾\xi \left(t\right)⩾{\xi }^{-}$, i.e., $\xi \left(t\right)$ decay phase

$$\begin{array}{ccc}\hfill u\left(t\right)& =\xi \left(t\right)+h+1-\sqrt{{(\xi \left(t\right)+h+1)}^2-e^{2\xi \left(t\right)}},\hfill & \hfill \mathrm{for}\mathrm{prey}\end{array}$$

(27a)

$$\begin{array}{ccc}\hfill v\left(t\right)& =\xi \left(t\right)+h+1+\sqrt{{(\xi \left(t\right)+h+1)}^2-e^{2\xi \left(t\right)}}.\hfill & \hfill \mathrm{for}\mathrm{predators}\end{array}$$

(27b)

with $t^{-1}\left(\xi \right)\to \xi \left(t\right)$ derived from (25b) together with $\xi \left(t^{\ast }\right)={\xi }^{+}\left(h\right)$.

Figure 4 displays the graph of the uncoupled analytic solutions for the time evolution of $u\left(t\right)$ and $v\left(t\right)$ when their respective growth and decay rates have equal magnitude and when the system’s energy is $h=2$. Remarkably, due to the uncoupling in (23), the $u\left(t\right)$ and $v\left(t\right)$ solutions do not explicitly depend on $\eta \left(t\right)$.

It is observed that the prey population $u\left(t\right)$ exhibits an initial growth at a rate significantly slower than its own fast decay rate, while the opposite is the case for the predators; also the peak population of the prey occurs when the population is mature, i.e., when the predator population is relatively small, $v\simeq 1$, and vice versa.

3.3. Case $\lambda \ne 1$

In the general case when $\lambda \ne 1$ the relationship between $G_{\lambda }\left(\eta \right)$ and its derivative $G_{\lambda }^{\prime }\left(\eta \right)$ is obtained by observing that

$$G_{\lambda }^{\prime }\left(\eta \right)={\displaystyle \frac{e^{\frac{\eta }{\lambda }}-e^{-\lambda \eta }}{\lambda +\frac{1}{\lambda }}}\mathrm{with}{G}_{\lambda }^{\prime }\left(\eta \right)=-G_{1/\lambda }^{\prime }(-\eta )(\lambda -\mathit{invariance})$$

(28)

Upon eliminating $\eta$ between (11) and (28), an implicit nonlinear 1st-order ODE relating G to its derivative $G^{\prime }$ is derived (for clarity the index $\lambda$ is omitted in the remainder of this section):

$${\left(G+{\displaystyle \frac{1}{\lambda }}{G}^{\prime }\right)}^{\lambda }{(G-\lambda G^{\prime })}^{1/\lambda }=1$$

(29)

Equation (29) is completely invariant under the change $\lambda \to -1/\lambda$ or equivalently changing $\lambda \to 1/\lambda$ together with $G^{\prime }\to -G^{\prime }$. As a result, similar to (22), in the $G-G^{\prime }$ phase space, (29) represents the positive and negative branches of a “skewed” hyperbola with orthogonal asymptotes, respectively $G^{\prime }=G/\lambda$ and $G^{\prime }=-\lambda G$, together with a vertex G′ = 0 located at $G=1$. For any value of $\lambda$, the function $G^{\prime }\left(\eta \right)$ reaches its extremes at the two roots of $G\left(\eta \right)=e^h$. Also, as expected, in the case of $\lambda =1$, (29) identically reduces to (22).

Being implicit, (29) can generally not be solved for $G^{\prime }$ as a function of G by standard algebraic techniques. A practical yet accurate approximation for the function $G^{\prime }\left(G\right)$ predicated on (20), which removes the $\eta$ dependence in (17b) and uncouples the system, is proposed below.

For the positive branch $G^{\prime }⩾0$, for large G, the function $G^{\prime }$ is asymptotic to $G^{\prime }=G/\lambda$. Equation (29) is first reformulated as

$$\lambda {\displaystyle \frac{G^{\prime }}{G}}=1-{\displaystyle \frac{1}{G^{{\lambda }^2+1}{\left(1+\frac{1}{\lambda }\frac{G^{\prime }}{G}\right)}^{{\lambda }^2}}}$$

(30)

The factor in parentheses in the denominator always satisfies the inequality

$${\left(1+{\displaystyle \frac{1}{\lambda }}{\displaystyle \frac{G^{\prime }}{G}}\right)}^{{\lambda }^2}<e^{\lambda {\displaystyle \frac{G^{\prime }}{G}}}$$

(31)

Upon approximating this factor by its exponential limit, (30) becomes

$$e^{\lambda {\displaystyle \frac{G^{\prime }}{G}}}\left(1-\lambda {\displaystyle \frac{G^{\prime }}{G}}\right)\cong {\displaystyle \frac{1}{G^{{\lambda }^2+1}}}$$

(32)

Since the G function is bounded by $e^h$, the right-hand side of (32) satisfies the following inequalities:

$$e^{-h({\lambda }^2+1)}⩽{\displaystyle \frac{1}{G^{{\lambda }^2+1}}}⩽1$$

(33)

Consistency between (32) and (33) requires the left-hand side of (32) to at most be of order O(1). Consequently, a Taylor expansion of the exponential function to 1st order can be used, yielding an explicit approximation for $G^{\prime }\left(G\right)$. (The neglected term is the quadratic term $s={(\lambda G^{\prime }/G)}^2/2$. To estimate the truncation error, the relevant range for G is the compact interval $[1,e^h]$, where the last bound corresponds to $\xi =-h$, i.e., when $\eta$ is either minimum or maximum. Near the vertex $G=1$, $G^{\prime }=0$ and $s\simeq 2{\lambda }^2(G-1)$, and the error is second order and small; at the other extremity $e^h$, defining the magnitude of the gap between the positive branch of $G^{\prime }$ and its asymptote $G/\lambda$ as $ϵ$ with $ϵ\ll 1$ and, if $\lambda e^{-h}\lesssim O\left(1\right)$, then the neglected term is $s={\displaystyle \frac{1}{2}}-\lambda e^{-h}ϵ$; s approaches its extreme value there and diminishes as the gap increases towards the vertex of G. This is consistent with the visible departure between the approximate and RK4 curves of Figure 5.) The negative branch $G^{\prime }⩽0$ is obtained from the positive branch G′ ⩾ 0 by $\lambda$ invariance

$$\begin{array}{cc}\hfill G^{\prime }\left(G\right)& \cong {\displaystyle \frac{G}{\lambda }}{\left(1-{\displaystyle \frac{1}{G^{{\lambda }^2+1}}}\right)}^{1/2}(\mathrm{positive}\mathrm{branch}{G}^{\prime }⩾0)\hfill \end{array}$$

(34a)

$$\begin{array}{cc}\hfill G^{\prime }\left(G\right)& \cong -\lambda G{\left(1-{\displaystyle \frac{1}{G^{1/{\lambda }^2+1}}}\right)}^{1/2}(\mathrm{negative}\mathrm{branch}{G}^{\prime }⩽0)\hfill \end{array}$$

(34b)

Remarkably, the above approximate function $G^{\prime }\left(G\right)$ satisfies the following three basic properties that are identical to those of an exact numerical solution of (29):

1.

At its vertex, when $G=1$, the function $G^{\prime }\left(G\right)$ reaches $G^{\prime }=0$;

2.

For $G\gg 1$, as expected, the positive branch of the function $G^{\prime }\left(G\right)$ is asymptotic to $G^{\prime }=G/\lambda$, whereas the negative branch is asymptotic to $G^{\prime }=-\lambda G$;

3.

For $\lambda =1$, the function $G^{\prime }\left(G\right)$ reduces to the exact predicate expression (20).

In the $G-G^{\prime }$ phase space, the explicit expressions (34) represent approximate positive and negative branches of the “skewed” hyperbola, defined by (29), bounded by the same orthogonal asymptotes. Graphic comparison between representations of (34) and a numerical solution of (29) for $G^{\prime }\left(G\right)$ shows quite reasonable agreement consistent with the first two properties of (34), particularly for the positive $G^{\prime }\left(G\right)$ branch when $\lambda ⩾1$ and conversely for the negative branch when $\lambda ⩽1$. For $\lambda \gg 1$ the graph of (34) exhibits two branches bounded by their respective orthogonal asymptotes, with the accuracy of this approximation increasing with increasing $\lambda$.

As intended, approximation (34) effectively uncouples the system (17) by explicitly removing the dependency on $\eta$ in the original ODE (17b): upon inserting the conservation Equation (12) into (34), Equation (17b) is replaced by a pair of two $\lambda$-invariant 1st-order autonomous nonlinear ODEs for $\xi \left(t\right)$ corresponding to each $\eta$ branch of the $\xi -\eta$ closed-form mapping

$$\begin{array}{cc}\hfill \dot{\xi }& =-{\displaystyle \frac{h+1+\xi }{\lambda }}{\left(1-{\displaystyle \frac{e^{\xi ({\lambda }^2+1)}}{{(h+1+\xi )}^{({\lambda }^2+1)}}}\right)}^{1/2}(\mathrm{positive}\eta -\mathrm{branch}\text{:}\eta ⩾0)\hfill \end{array}$$

(35a)

$$\begin{array}{cc}\hfill \dot{\xi }& =\lambda (h+1+\xi ){\left(1-{\displaystyle \frac{e^{\xi (1/{\lambda }^2+1)}}{{(h+1+\xi )}^{(1/{\lambda }^2+1)}}}\right)}^{1/2}(\mathrm{negative}\eta -\mathrm{branch}\text{:}\eta ⩽0)\hfill \end{array}$$

(35b)

As $\lambda \to 1$ the approximation (34) approaches the exact solution (20), and, for $\lambda =1$, the two branches of (23b) are strictly recovered. Upon choosing the time $t=0$ when $\xi \left(0\right)={\xi }^{-}\left(h\right)$, a complete oscillation period is obtained by integration over the corresponding negative $\eta$ branch in (35b) until $\xi \left(t\right)$ reaches ${\xi }^{+}\left(h\right)$, followed by an integration over the positive $\eta$ branch (35a) until ${\xi }^{-}\left(h\right)$ is reached. Even though $\xi \left(t\right)$ is not explicitly expressed as a function of time t, the arbitrary $\lambda \ne 1$ problem has thus been reduced to a pair of simple quadratures for the function $t\left(\xi \right)$. The approximate function $t\left(\xi \right)$ is monotonic and continuous on the respective integration intervals ${\xi }^{-}⩽\xi ⩽{\xi }^{+}$ and ${\xi }^{+}⩾\xi ⩾{\xi }^{-}$, and its inverse function $\xi \left(t\right)$ exists and is unique, monotonic, and continuous on each interval.

$$\begin{array}{cc}\hfill t\left(\xi \right)& ={\int }_{{\xi }^{-}}^{\xi }{\displaystyle \frac{1}{\lambda (h+1+x)}}{\left(1-{\displaystyle \frac{e^{x(1/{\lambda }^2+1)}}{{(h+1+x)}^{(1/{\lambda }^2+1)}}}\right)}^{-1/2}dx(\mathrm{negative}\eta -\mathrm{branch})\hfill \end{array}$$

(36a)

$$\begin{array}{cc}\hfill t\left(\xi \right)& =-{\int }_{{\xi }^{+}}^{\xi }{\displaystyle \frac{\lambda }{h+1+x}}{\left(1-{\displaystyle \frac{e^{x({\lambda }^2+1)}}{{(h+1+x)}^{({\lambda }^2+1)}}}\right)}^{-1/2}dx(\mathrm{positive}\eta -\mathrm{branch})\hfill \end{array}$$

(36b)

The LV problem is then completed for the function $\eta \left(t\right)$ by directly integrating the linear Equation (17a) through standard numerical techniques.

To assess the accuracy of the uncoupled approximate solution (35), a comparison is made with the exact numerical solutions of the original coupled LV system (17). Upon using the respective values $\lambda =2$ and $h=2$, identical to those of Figure 2, Figure 5 presents the functions $\xi \left(t\right)$ and $\eta \left(t\right)$, obtained, respectively, by numerically integrating (35) and (17) simultaneously through a standard 4th-order RK4 method. Despite not being exact solutions, the ODEs (35) provide a reasonably accurate solution for both functions $\xi \left(t\right)$ and $\eta \left(t\right)$ over an entire period. However, when $\lambda >1$, an underestimation of the time taken to reach ${\xi }^{+}\left(h\right)$ is compensated by an overestimation of the time to reach ${\xi }^{-}\left(h\right)$.

As expected, the accuracy of the solutions obtained with approximation (35) increases with increasing $\lambda$. This can be observed by measuring the mismatch $\mathrm{\Delta }{t}^{\ast }$ between the exact $\xi \left(t\right)$ peak amplitude time $t^{\ast }$ and the approximate time $t_a^{\ast }$, obtained, respectively, with the RK4 solution and the approximation (35). In Figure 5, for an energy $h=2$ and for $\lambda =2$, this mismatch is $\mathrm{\Delta }{t}^{\ast }=0.197$ or 5.80% of $t^{\ast }$.

Table 1. Peak amplitude, period, and relative mismatch between the RK4 solution and the approximate solution (35) as a function of $\lambda$ for $h=2$.

$\lambda$	1.5	2.0	2.5	3.0	4.0	5.0
$t_{\lambda }^{\ast }\left(h\right)$	4.09	3.40	2.94	2.63	2.17	1.88
$T_{\lambda }\left(h\right)$	11.53	12.89	14.50	16.36	20.23	24.28
$\mathrm{\Delta }{t}^{\ast }(\%)$	5.94%	5.80%	5.02%	4.49%	3.70%	3.21%

displays the peak amplitude $t_{\lambda }^{\ast }\left(h\right)$ together with the period $T_{\lambda }\left(h\right)$ and the relative mismatch $\mathrm{\Delta }{t}^{\ast }(\%)$ as a function of $\lambda$ for $h=2$. It demonstrates the accuracy improvement of the approximation (35): for a given h, as $\lambda$ increases, the period increases smoothly, whereas the time $t^{\ast }\left(2\right)$ simultaneously diminishes, implying a faster relative growth of $\xi \left(t\right)$. A similar analysis for $\lambda =2$ and various values of h shows that the approximation $\eta \left(t\right)$ stays within 4% of the RK4 solution, but it diverges beyond $h\simeq 4$ since the positive $\eta$ branch of the approximate function fails to return to 0 in the vicinity of the time $t=T_2\left(h\right)$.

Remarkably, in the high-energy limit $(h\gg 1)$, upon keeping the leading term of the approximate system in (35), the asymptotic behavior of the LV system is modeled as a system of two coupled linear 1st-order ODEs for $\xi \left(t\right)$ and $\eta \left(t\right)$. In this asymptotic limit, together with the linear ODE (17a) for $\eta \left(t\right)$, the system admits trivial exponential solutions. For example, the asymptotic solutions $(h\gg 1)$ for the growth phase $({\xi }^{-}⩽\xi ⩽{\xi }^{+})$ simply are

$$\begin{array}{cc}\hfill \xi \left(t\right)& =e^{{\xi }^{-}+\lambda t}-(h+1)\hfill \end{array}$$

(37a)

$$\begin{array}{cc}\hfill \eta \left(t\right)& ={\displaystyle \frac{1}{\lambda }}\left(\xi \left(t\right)-{\xi }^{-}\left(h\right)\right)-t\hfill \end{array}$$

(37b)

The decay phase asymptotic solutions for $\xi \left(t\right)$ are obtained by $\lambda$ invariance, namely $\lambda \to -1/\lambda$ together with ${\xi }^{-}\left(h\right)\to {\xi }^{+}\left(h\right)$.

Lastly, upon inserting the hybrid functions $\xi \left(t\right)$ and $\eta \left(t\right)$ derived from (36) into the definition (5) of the prey and predator species, the respective standard solutions for the original populations $u\left(t\right)$ and $v\left(t\right)$ are fully recovered,

$$\begin{array}{cc}\hfill u\left(t\right)& =e^{\xi \left(t\right)-\lambda \eta \left(t\right)}\mathrm{for}\mathrm{prey}\hfill \end{array}$$

(38a)

$$\begin{array}{cc}\hfill v\left(t\right)& =e^{\xi \left(t\right)+\eta \left(t\right)/\lambda }\mathrm{for}\mathrm{predators}\hfill \end{array}$$

(38b)

3.4. Ranges of $\lambda$ and h

The oscillation amplitude ${\xi }^{+}\left(h\right)-{\xi }^{-}\left(h\right)$ is mostly associated with the energy h, whereas the respective growth and decay rates $\lambda$ and $1/\lambda$ together with the time $t_{\lambda }^{\ast }$ for peak amplitude of $\xi \left(t\right)$ are mostly related to the magnitude of the coupling ratio $\lambda$.

Further, the $\lambda$ and h ranges for which the approximation (35) is quantitatively acceptable are directly related to the uncoupling of $G^{\prime }\left(G\right)$ since, when $\lambda >1$, it mostly affects the values of the positive $\eta$ branch, particularly in the vicinity of $T_{\lambda }\left(h\right)$. Hence, as seen above, the energy range is limited to moderate values $h⩽4$. It should be noted that, in the case of $\lambda =1$, there is no limit on h, as shown in Section 4. Since the coupling ratio mostly affects the growth and decay rates of $\xi \left(t\right)$ in which shorter approximate times $t_{\lambda }^{\ast }\left(h\right)$ in the growth phase compensate for much longer approximate times during the decay phase, the resulting sum $T_{\lambda }\left(h\right)$ becomes a smoothly increasing function of $\lambda$, as shown in Table 1. Consequently, a moderate limit $\lambda ⩽5$ is an acceptable range for the coupling ratio. From an ecological viewpoint, this is likely to be a conservative high limit since it implies $\alpha =25\delta$, a very high growth rate for prey relative to the predator decay rate.

4. Oscillation Period of the LV System

The unique $\lambda$ invariance property of ${\eta }^{\pm }(\xi ,\lambda )$ in (15) directly enables establishing two important properties of the LV system period. First, the author has shown that, for any value of the positive orbital energy h, the LV system oscillation periods respectively corresponding to the coupling ratio $\lambda$ and its inverse $1/\lambda$ are equal [7]. This is evident when observing the symmetry of the close-orbit mapping for $\lambda$ and $1/\lambda$ in Figure 1.

$$T_{\lambda }\left(h\right)=T_{1/\lambda }\left(h\right)$$

(39)

Second, the author has also established that, for any value of the positive orbital energy h, the LV system oscillation period $T_{\lambda }\left(h\right)$ is an increasing function of $\lambda$ for $\lambda ⩾1$ (decreasing for 0 < $\lambda ⩽1$), and the period is shortest for $\lambda =1$. This is clearly displayed in Figure 6.

Consequently, an exact regular expression for the nonlinear LV system oscillation period, valid for any value of the coupling ratio $\lambda ⩾1$ and any value of the orbital energy h, is directly derived from (19) as an integral over the two branches of the $\xi$ − $\eta$ closed-form mapping

$$\begin{array}{c}{T}_{\lambda }\left(h\right)={\displaystyle \frac{1}{\sqrt{\alpha \delta }}}{\displaystyle \frac{\left({\eta }^{-}(-h,\lambda )-{\eta }^{+}(-h,\lambda )\right)({\xi }^{+}-{\xi }^{-})}{(h+{\xi }^{+})(h+{\xi }^{-})}}+\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\displaystyle \frac{1}{\sqrt{\alpha \delta }}}{\int }_{{\xi }^{-}}^{{\xi }^{+}}{\displaystyle \frac{{\eta }^{-}(x,\lambda )-{\eta }^{-}(-h,\lambda )+{\eta }^{+}(-h,\lambda )-{\eta }^{+}(x,\lambda )}{{(h+x)}^2}}dx\end{array}$$

(40)

As in (18), the convergence of (40) at $x=-h$ is ensured since the numerator is simply the difference of two functions $f^{\pm }\left(x\right)$ introduced in the Paragraph under Formula (18).

4.1. Case $\lambda \ne 1$

In this case the exact LV oscillation period $T_{\lambda }\left(h\right)$ is obtained by numerically solving the ODE system (17) as done for Figure 2. For each value of the coupling ratio $\lambda$, the period $T_{\lambda }\left(h\right)$ is then uniquely expressed in terms of the dimensionless LV energy functions ${\mathrm{\Theta }}_{\lambda }\left(h\right)$,

$$T_{\lambda }\left(h\right)={\displaystyle \frac{2\pi }{\sqrt{\alpha \delta }}}{\mathrm{\Theta }}_{\lambda }\left(h\right)$$

(41)

As shown in Figure 6, for any value of the coupling ratio $\lambda$, each function ${\mathrm{\Theta }}_{\lambda }\left(h\right)$ (and by extension the period $T_{\lambda }\left(h\right)$) is a monotonically increasing function of the system’s energy h [20]. Also, for any value of $\lambda$ or h, ${\mathrm{\Theta }}_1\left(h\right)<{\mathrm{\Theta }}_{\lambda }\left(h\right)$ [7]. In this general case, an asymptotic formula for the LV system oscillation period $T_{\lambda }\left(h\right)$, valid at high energy $(h\gg 1)$, is obtained from the asymptotic solutions (38). The contribution $T_{\lambda }^{+}\left(h\right)$ of the exponential growth phase of $\xi \left(t\right)$ to the period is readily obtained from (38a) since $\eta \left(t\right)=0$ when $\xi \left(t\right)$ reaches its maximum ${\xi }^{+}\left(h\right)$; the contribution $T_{\lambda }^{-}\left(h\right)$ of the decay phase is obtained by $\lambda$ invariance. As a result, an asymptotic expression for the LV system period $T_{\lambda }\left(h\right)$ simply becomes proportional to the sum of the $\xi \left(t\right)$ function growth and decay rates, $\lambda$ and $1/\lambda$, respectively. In this asymptotic limit the period is proportional to the factor $\lambda +1/\lambda$, which increases with $\lambda$, and to the oscillation amplitude ${\xi }^{+}\left(h\right)-{\xi }^{-}\left(h\right)$, which is an increasing function of h, consistent with Rothe’s remark that LV system oscillations with larger amplitudes are slower [21]:

$$T_{\lambda }\left(h\right)\cong {\displaystyle \frac{\pi }{\sqrt{\alpha \delta }}}\left(\lambda +{\displaystyle \frac{1}{\lambda }}\right)\left({\xi }^{+}\left(h\right)-{\xi }^{-}\left(h\right)\right)$$

(42)

Remarkably, this asymptotic expression provides reasonably good estimates of the values of ${\mathrm{\Theta }}_{\lambda }\left(h\right)$ at high energies, as can be verified for $h=5$ and $\lambda =5$, for which (42) predicts ${\mathrm{\Theta }}_5\left(5\right)=6.69$ when the exact value is 6.76.

Waldvogel has shown that the LV period grows monotonically with the energy. Figure 6, obtained from numerical integration, shows that the dimensionless energy function is a monotonically increasing function of the energy and that ${\mathrm{\Theta }}_{\lambda }\left(h\right)>{\mathrm{\Theta }}_1\left(h\right)$, implying that, for a given energy h, higher values of $\lambda$ correspond to higher values of the period in agreement with [20]. The function ${\mathrm{\Theta }}_1\left(h\right)$ tabulated in Table 1 is a monotonically increasing function of h and so is the asymptotic limit (42). A comparison between Waldvogel’s Taylor expansion of the period $T\left(h\right)$ for small energies with ${\mathrm{\Theta }}_1\left(h\right)$ derived by expansion of a complete elliptic solution for $h\ll 1$ is made in the next section.

Upon comparing the methods of Volterra [1], Hsu [22], Waldvogel [20], and Rothe [21], Shih demonstrated that all of their integral representations for the period of the two-species LV system are equivalent to his own solution in terms of a sum of several convolution integrals [23]. Subsequent approximations in terms of power series [24] or perturbation expansions [25] have also been published. The exact solution (40) does not rely on a convolution and only depends on a simple integral over the two branches of the $\xi$ − $\eta$ mapping.

4.2. Case $\lambda =1$

In this particular case, the exact LV system period $T_1\left(h\right)$ is uniquely expressed in terms of a dimensionless energy function ${\mathrm{\Theta }}_1\left(h\right)$ as

$$T_1\left(h\right)={\displaystyle \frac{2\pi }{\alpha }}{\mathrm{\Theta }}_1\left(h\right).$$

(43)

The energy function ${\mathrm{\Theta }}_1\left(h\right)$ introduced in [7] is defined by integrating (25a) over the entire $\xi$ interval,

$${\mathrm{\Theta }}_1\left(h\right)={\displaystyle \frac{1}{\pi }}{\int }_{{\xi }^{-}}^{{\xi }^{+}}{\displaystyle \frac{dx}{\sqrt{{(x+h+1)}^2-e^{2x}}}}.$$

(44)

An integral representation in terms of the Lambert W function is given in Appendix A (A6). At small orbital energy ($h\ll 1$) where ${\xi }^{\pm }\left(h\right)\simeq \pm \sqrt{2h}$, the function ${\mathrm{\Theta }}_1\left(h\right)$ is directly expressed in terms of the complete elliptic integral of the first kind $\mathit{K}\left(k\right)$ with modulus k [7]. A standard series expansion for $\mathit{K}\left(k\right)$ yields

$${\mathrm{\Theta }}_1\left(h\right)=1+{\displaystyle \frac{1}{3}}h+{\displaystyle \frac{1}{42}}{h}^2+O\left(h^3\right).$$

(45)

For small oscillation amplitudes, the integral (44) becomes independent of the energy h and is exactly equal to $\pi$, so ${\mathrm{\Theta }}_1\left(h\right)=1$ in (43); the LV system becomes that of two coupled harmonic oscillators for which the period $T_1\left(h\right)$ solely depends on the pulsation $\alpha$, as already established [1,20].

Waldvogel [20] has derived a convergent expansion of the period $T\left(h\right)$ for sufficiently small values of h. Defining $T\left(h\right)={\displaystyle \frac{2\pi }{\sqrt{\alpha \delta }}}{\mathrm{\Theta }}_w\left(h\right)$ introduces the expression for the Waldvogel energy function ${\mathrm{\Theta }}_w\left(h\right)$, which can be rearranged as

$${\mathrm{\Theta }}_w\left(h\right)=1+{\displaystyle \frac{1}{6}}({\lambda }^2+{\displaystyle \frac{1}{{\lambda }^2}})h+{\displaystyle \frac{1}{144}}{({\lambda }^2+{\displaystyle \frac{1}{{\lambda }^2}})}^2{h}^2+O\left(h^3\right).$$

(46)

In the case $\lambda =1$ this expansion becomes

$${\mathrm{\Theta }}_w\left(h\right)=1+{\displaystyle \frac{1}{3}}h+{\displaystyle \frac{1}{36}}{h}^2+O\left(h^3\right).$$

(47)

The series expansion (45) is in agreement with that of Waldvogel to 1st order in h and differs at 2nd order by a very small amount of 3.96 ${10}^{-3}{h}^2$, presumably because the term of $O\left(h^3\right)$ in (47) is negative while that of (45) is positive. Overall the agreement is quantitatively reasonable, justifying the use of (44) and (A5), at least for small energies.

At high orbital energy ($h\gg 1$), the contribution from the exponential term in (44) becomes negligible since $\xi <0$ over most of the integration interval except when $\xi$ approaches ${\xi }^{+}\left(h\right)$. By definition, $\xi \left(t\right)⩾{\xi }^{-}\left(h\right)$. Approximating the exponential term by its lowest value $e^{2{\xi }^{-}\left(h\right)}$ and performing the integration yields a useful closed-form asymptotic expression for ${\mathrm{\Theta }}_1\left(h\right)$,

$${\mathrm{\Theta }}_{1,\mathrm{asym}}\left(h\right)\cong {\displaystyle \frac{1}{\pi }}\left({\xi }^{+}\left(h\right)-{\xi }^{-}\left(h\right)+ln\left(2\right)\right).\mathrm{with}h\gg 1$$

(48)

The dimensionless function ${\mathrm{\Theta }}_1\left(h\right)$ presented in

Table 2. Roots of $e^{\xi }-\xi -1=h$ as a function of the energy h from (16) and values of ${\mathrm{\Theta }}_1\left(h\right)$ from (44).

h	0.3	0.5	1	2	3	5	7	10
${\xi }^{-}\left(h\right)$	−0.889	−1.198	−1.841	−2.948	−3.981	−5.998	−8.000	−11.00
${\xi }^{+}\left(h\right)$	0.686	0.858	1.146	1.505	1.749	2.091	2.336	2.611
${\mathrm{\Theta }}_1\left(h\right)$	1.102	1.173	1.355	1.728	2.102	2.828	3.535	4.569

for various values of the energy h is also displayed in Figure 6: it shows that ${\mathrm{\Theta }}_1\left(h\right)$ is a monotonically increasing function of the energy-dependent amplitude ${\xi }^{+}\left(h\right)-{\xi }^{-}\left(h\right)$. Also shown is the asymptotic approximation (48), which is practically indistinguishable from the exact function ${\mathrm{\Theta }}_1\left(h\right)$ for $h⩾4$.

5. Conclusions

The objective of this research has been to formulate the LV problem into a simple compact form while attempting to uncouple the basic ODE system: the historic LV-coupled first-order nonlinear ODE system of two interacting species has been recast in terms of a single positive coupling parameter $\lambda$, the ratio of the relative growth/decay rates of each species taken independently. A Hamiltonian formulation combined with a linear transformation introducing hybrid variables partially uncouples the system into a $\lambda$-invariant set of two first-order ODEs, with one being linear. This enables $t\left(\xi \right)$ to be formulated in terms of a simple integral. As a result, an exact quadrature solution of the LV problem is derived for any value of the coupling ratio $\lambda$ and the system’s energy.

In the general case of $\lambda \ne 1$, an attempt has been made to uncouple the LV system by deriving an approximate ODE for $t\left(\xi \right)$ that is autonomous, like in the case of $\lambda =1$. This has provided accurate practical approximate integral solutions that are not solvable in terms of standard functions. The quantitatively acceptable approximation ranges for the energy h and the coupling ratio $\lambda$ are provided. Remarkably, at high orbital energies, the uncoupled system becomes entirely linear, admitting trivial closed-form asymptotic exponential solutions. Further, for any value of the coupling ratio $\lambda$, the oscillation period is shown to be a monotonically increasing function of the energy and of $\lambda$; at high energies and $\lambda \gg 1$, a trivial reasonably accurate asymptotic expression for the oscillation period is derived, showing that higher oscillation amplitudes result in longer oscillation periods.

The $\lambda =1$ case, in which the intrinsic exponential growth and decay rates of the predators and prey are equal, represents a genuine, if special, ecological symmetry condition: it is the condition under which $\lambda$ invariance becomes a self-duality. As such, through the exact uncoupling (22), together with the simple quadrature (25) expressed in terms of a bounded integral through the Lambert W function and the expansion of the energy function ${\mathrm{\Theta }}_1\left(h\right)$, this case represents a benchmark or limiting case for the accuracy of the $\lambda \ne 1$ approximation.

When $\lambda =1$, the LV problem completely uncouples and an exact explicit solution is derived in terms of the system’s orbital energy as a simple quadrature for the time evolution of one of the hybrid functions, the solution for the other function being explicitly expressed in terms of the former. Predicated on the exact closed-form energy-dependent turning-point solutions, this basic quadrature is restructured in terms of the Lambert W function, which eliminates the square root singularity but shifts the burden to the delicate exercise of accounting for the ill condition of the Lambert function near its branch point. Appendix A establishes that the integral is regular at the branch point but must be evaluated numerically. Also, exact uncoupled analytic solutions for each of the original prey and predator populations are derived as a function of $\xi \left(t\right)$. Lastly, an exact analytic expression for the LV system oscillation period is derived in terms of a dimensionless energy function together with a simple closed-form asymptotic expression for high energies.

This research highlights several areas that warrant further investigation: (1) not unlike several other authors, due to the particular nonlinearity in the coupling of the LV problem, compact solutions for the evolution of the predators and prey are expressed in terms of quadratures only except in the case of asymptotic solutions; (2) the $\lambda \ne 1$ uncoupling is only an approximation, yet it is reasonably accurate for moderate energy values; (3) even though predicated on the closed-form solutions for turning points, the Lambert W function representation does not yield a full solution in terms of standard functions, although it eliminates the square root singularity in an integral that poses a stiff numerical integration challenge.

Nevertheless the paper presents valuable contributions: (1) the Hamiltonian formulation for hybrid variables $\xi \left(t\right)$ and $\eta \left(t\right)$ and the $\lambda$ invariance${\eta }^{\pm }(\xi ,\lambda )=-{\eta }^{\mp }(\xi ,1/\lambda )$, as well as reducing the system to two ODEs, with one being linear (which is not the case in the LV system) and, in the case of $\lambda =1$, the second being autonomous, thereby providing structural insight into the LV problem; (2) exact closed-form solutions for the turning points are expressed in terms of the Lambert W function; (3) the numerical figures (RK4 vs. quadrature approximate solutions) are consistent with the stated accuracy; (4) the $\lambda =1$ case, which may be perceived as too narrow, particularly from an ecology standpoint even though it provides exact closed-form mapping and quadrature solutions, is shown to fold in the general $\lambda \ne 1$ approximate uncoupling case; (5) formulae for the period are expressed as simple integrals over the $\xi -\eta$ mapping and not as convolution integrals that are more difficult to evaluate numerically; (6) useful asymptotic closed-form expressions for $\xi \left(t\right),\eta \left(t\right)$, and $T_{\lambda }\left(h\right)$ are also derived.