The mathematical theory of endosymbiosis I

Nonlinear Analysis: Real World Applications 12 (2011) 3238–3251

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Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa

The mathematical theory of endosymbiosis I Peter L. Antonelli a,∗ , Solange F. Rutz b,c , Carlos E. Hirakawa d a

University of Alberta, Edmonton, AB, Canada

b

IBRAG, UERJ, Rio de Janeiro, RJ, Brazil

c

Department of Mathematics, Federal University of Pernambuco, Recife, PE, Brazil

d

IME, UERJ, Rio de Janeiro, RJ, Brazil

article

info

Article history: Received 10 May 2011 Accepted 21 May 2011 Keywords: Finsler and projective geometries KCC-theory Biological evolution Endosymbiosis Ancestral commune Berwald spaces Volterra–Hamilton systems

abstract This work presents a new model of the evolutionary process formulated by the Serial Endosymbiosis Theory represented by a succession of stages involving different metabolic and ecological interactions among populations of bacteria considering both the population dynamics and production processes of these populations. In such an approach we make use of systems of differential equations known as Volterra–Hamilton systems as well as some geometric concepts involving KCC Theory and the Projective Geometry of Berwald Spaces and also correct a statement of M. Matsumoto in the literature on this topic. We also recount in some detail previous work comparing production stability of Endosymbiosis Theory with that of Ancestral Commune Theory. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction According to the Ancestral Commune Theory of Carl Woese, life started as a loose conglomerate of many types of protocells, some 4000 MYBP. A billion years later trading loose bits of RNA and DNA between these precursors gave way to Darwinian natural selection. An anaerobic thermophile bacterium was present in the primordial soup and possessed a nuclear genome. This evolved prokaryote was subsequently parasitized by a bacterium that had a flagellum for swimming and could process oxygen for its energy. It is called a mitochondrian, while the symbiocosm is called a eukaryote. It is the common ancestor of all creatures except the prokaryotes themselves. This theory has been fully validated via molecular genetics and is credited to Lynn Margulis. It is therefore obvious that the most ancient symbiocosm is the eukaryote cell. For plant species, one has the chloroplasts as well as mitochondia, both of which occur in variable small numbers across the plant and animal species spectrum. Using the Volterra–Hamilton method we have compared the evolutionary theories of Woese and Margulis and because we treated both within a single logical framework, in spite of the fact that the communes of proto-cells in Woese’s theory predate the evolved eukaryote symbiocosm, sensible comparison was achieved. We are now going to recount the detailed mathematical material. Using the Volterra–Hamilton systems as a logical method we have compared (see [1] for recent advances in endosymbiosis) the evolutionary theories of Carl Woese and Lynn Margulis and found the former suffers from robust instability while the later is robustly stable in its production processes [2,3]. N i = density of i-th bacterial population, assumed to satisfy classical logistic dynamics dN i dt

∗

= λN i (1 − α(i) N i ),

Corresponding author. E-mail addresses: [email protected] (P.L. Antonelli), [email protected] (S.F. Rutz), [email protected] (C.E. Hirakawa).

1468-1218/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2011.05.023

P.L. Antonelli et al. / Nonlinear Analysis: Real World Applications 12 (2011) 3238–3251

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with pre-symbiont condition (all lambdas equal), i = 1, 2, . . . , n and repeated indices are summed except if there is a parenthesis. Since our model will describe ecology and chemical production (in the form of modular bits of RNA), we introduce the Volterra production equation dxi

= k(i) N i .

dt

We require that, for either model, our evolved system has the form d2 x i ds2

+ Gijk

dxj dxk ds ds

= 0,

where the n3 coefficients are constants or involve xi . This class of dynamical systems will encompass both the primeval and the evolved systems, and serve for modeling either Margulis’ or Woese’ theories. In order to accommodate our ergonomics, i.e., division of labour, we require the production parameter to be given by the cost of production functional, ds = F (x, dx) > 0. Moreover, F is to be positively homogeneous of degree 1 in dx = (dx1 , . . . , dxn ), that is, for any positive constant c,

 F

x, c

dx



dt

  dx = c F x, , dt

so that ds/dt, the rate of production in the symbiocosm, depends on individual bacterial rates dxi /dt through the cost t F (x, dx/dt ). The arc-length s = t F (x, dx/dt )dt represents the total production of the symbiocosm in the interval (t2 − t1) 0

along a given curve x(t ) = (x1 (t ), . . . , xn (t )). The homogeneity of F means that, if all the individual rates dxi /dt are magnified by a factor of c, then ds/dt is so magnified. This forces s to be independent of the time measure. We have to introduce the expression Hs = (1/2)F 2 (x, dx/dt ), which yields Euler–Lagrange equations. Note that Hs is therefore positively homogeneous of degree 2 in dx/dt, thus, multiplying each dxi /dt by a positive constant c implies that Hs is multiplied by c 2 . Furthermore, Hs defines two classes of systems, namely, the Riemannian class, where Hs is quadratic in dx/dt, and the Finsler (non-Riemannian) class, where Hs is not quadratic, but is homogeneous of degree 2 in dx/dt. Here are two examples, one for each class. For the former, quadratic case, we may have Hs =

1 2

e

2αi xi



dx1 dt

2

 + ··· +

dxn

2 

dt

,

while for the latter, non-quadratic but homogeneous case, we have Hs =

1 2

e2φ(x)

(dx2 /dt )2+2/λ , (dx1 /dt )2/λ

where we have taken n = 2, λ is a positive constant and φ(x) is an arbitrary polynomial on x1 and x2 . We will see in the following sections that the former applies to Woese’s theory, while the latter applies to Margulis’ theory. To solve the problem we need to use the techniques of Finsler geometry. Our main result is that, for Hs given as above, with φ(x) = −α1 x1 + (λ + 1)α2 x2 + ν3 x1 x2 , and λ > 0, αi > 0 and ν3 non-zero, the Euler–Lagrange equations are

 1 dy   + λ(α1 − ν3 x2 ).(y1 )2 = 0,  ds    dy2 ν3 1   + λ α2 + x .(y2 )2 = 0. ds λ+1

(1)

Note that, if ν3 = 0, then the original double logistic system is obtained.1 Moreover, Liapunov stability of this system is completely determined by the sign of the curvature (see Appendices): K =

λ2 ν3 λ+1



y1 y2

1+2/λ

. exp(−2[−α1 x1 + (λ + 1)α2 x2 + ν3 x1 x2 ]).

If ν3 > 0, then stability results, while the reverse is true for ν3 < 0. Geodesics of 2-dimension positively curved spaces, such as a sphere, will remain close generally in x-space, the system being, therefore, stable, while for those with a negative or zero curvature, as a trumpet surface or a plane, respectively, will not, yielding unstable systems. The parameter ν3 is called the exchange parameter. Thus, Margulis # 1 has stable production. Index # 1 indicates the parasite and # 2, the host [4–7]. In the Ancestral Commune model we disallow explicit xi in the coefficients, but allow the number of species to be large. This is our model of a loose conglomerate of diverse bacterial species. Neither do we allow the coefficients to depend on

1 Where ds = eλt dt must be employed to transform to real time t from parameter s.

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the population sizes, as this would be inclusion of social interactions. As before, we will try for a quadratic cost functional as the simplest possible. Using the fundamental theorem of Volterra–Hamilton systems to arrive at the equation above as the n-dimensional functional, using the additional assumption that our community is simple. This means precisely that each bacterial species in the conglomerate exchanges chemical information with at least one other. If this were not assumed, then the fundamental theorem ensures that the conglomerate splits into simple sub-communities, each of the above type, i.e., having the previously stated, quadratic Hs , as cost functional, with n varying from one community to another, totaling the original number of species. The dynamic equations which are generated by the cost functional through the calculus of variations are actually Euler–Lagrange equations, with coefficients Gijk as Giii = αi , Giji

Gijk = 0,

= αj ,

Giij

=

Gijj

= −αi ,

i ̸= j ̸= k

i ̸= j

(2)

i ̸= j.

These coefficients of interaction completely characterize our model of Ancestral Commune. It is especially significant that the Gijk , with all indexes different, vanishes. For example, for a 3 species conglomerate, G123 , G213 , G312 all vanish, which means there are no higher-order interactions, so that species 2 and 3 do not have an interaction which influences species 1, etc. This is a consequence of our model. It is consistent with the loose conglomerate of Woese. Another consequence is that the scalar curvature R is given by R = −(n − 1)(n − 2) exp(−2αi xi )[(α1 )2 + · · · + (αn )2 ]. As one can notice from the above equation, R increases quadratically with n (all other quantities being constant), becoming ever more negative, and so unstable, the larger the commune. The Riemann scalar curvature R and Berwald’s Gaussian curvature K play an important role in stochastic problems in curved spaces. The stochastic version of the above theory comparison has been published [8]. The results are essentially the same: Woese’s theory suffers from instability in the chemical exchanges while Margulis’ does not. We had to employ the theory of noise in Finsler spaces due to Antonelli and Zastawniak [9]. Representing the Serial Endosymbiosis Theory by a succession of stages, let us consider the initial stage, the time when a bacterium parasitizes a prokaryote. A simple model, based on the Volterra–Hamilton system for this stage, can be given considering Giii = αi ,

Gijj = 0,

Giij = Giji = −βi ,

j Gij

i ̸= j, αi > 0,

= Gjji = βj ,

i ̸= j, βi , βj > 0,

with i, j = 1, 2. We will search for a model for the change of state between parasitism and endosymbiosis. To do so, we will use the biological concept of heterochrony and mathematical concepts from projective geometry. Heterochrony, considered an important evolutionary process, is defined as a time-sequencing change in the ontogenetic process, which describes the origin and development of the individual from the fertilized egg to its adult form. In other words, heterochrony can be understood as evolutionary changes in an individual caused by changes in the pace of development. In our context, we will use this concept as an analogy for the evolutionary process of the Margulis theory in which the evolutionary transition between stages occurs haphazardly. For the modeling of heterochrony we will use the projective geometry, implying a time-sequencing change in the parameter t between these stages. 2. Projective geometry 2.1. Local sprays Consider a smooth connected n-manifold M n and select a trivializing chart (U , h) on M n for the slit tangent bundle T˜ M n (i.e. with the zero section removed). A (local) spray G in (U , h) is a system of ode’s d2 x i ds2

  dx + 2Gi x, = 0, ds

(i = 1, . . . , n),

(3)

where the n functions Gi are C oo on U in xi , . . . , xn and in dx1 /ds, . . . , dxn /ds (off the zero section), are otherwise continuous and are second-degree positively homogeneous in the dxi /ds.2 The path parameter s is special. For a general parameter t along solutions of Eq. (3) we have s′′ x¨ i + 2Gi (x, x˙ ) = ′ x˙ i , s where s′ := ds/dt , s′′ := (s′ )′ , x˙ i := dxi /dt and x¨ i := d2 xi /dt 2 . 2 In Appendix A, sprays require ϵ i = 0, otherwise we have only semisprays.

(4)

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Consider ψ(x, x˙ ), a smooth scalar function on T˜ M n , which is first-degree positively homogeneous in x˙ 1 , . . . , x˙ n . The quantities x¨ i + 2Gi

x¨ j + 2Gj

, ∀i, j ∈ {1, . . . , n} x˙ i x˙ j remain unchanged by the transformation =

¯ i := Gi + ψ.˙xi Gi → G

(5)

¯ in (U , h). That is, there exists a diffeomorphism which smoothly maps solutions which sends the spray G in (U , h) to spray G ¯ Such a mapping is called the projective transformation of G onto G¯ in (U , h). of G into solutions of G. One obtains from the spray parameter s (i.e. one which makes the RHS of Eq. (4) vanish) a new spray parameter determined by ψ . Namely,

∫

s¯ = A + B.

e

2/(n+1) γ ψ(x,dx/dt˜)dt˜ ds



,

(6)

where t˜ is any parameter along any path γ , that is a solution of G, and A, B are constants of integration. We can see the effect of this projective change, or time-sequencing change, by considering the canonical spray connection coefficients in (U , h): Gij := ∂˙j Gi ,

Gijk := ∂˙k Gij ,

(7)

where ∂˙l indicates partial differentiation with respect to x˙ . The transformation of coordinates from (U , h) to (U¯ , h¯ ), i.e. from x1 , . . . , xn to x¯ 1 , . . . , x¯ n , has the effect [10], l

∂ x¯ r ∂ x¯ s ¯ i ∂ x¯ i r ∂ 2 x¯ i G = G − . rs jk ∂ xj ∂ xk ∂ xr ∂ xj ∂ xk

(8)

Because Gi are homogeneous of the second degree in x˙ l , we have the equivalent expression for Eq. (3) d2 x i ds2

+

Gijk



x,

dx



ds

dxj dxk ds ds

= 0.

(9)

Upon time-sequencing change ψ of Eq. (9), we have by differentiation in (U , h)

¯ ijk = Gijk + δji ψk + δki ψj + x˙ i ∂˙k ψj , G

(10)

where ψl = ∂˙l ψ . Note that the 3-index G’s are the local coefficients of the Berwald connection (see Appendix B) or [11] where they are called spray connection coefficients. Define 1

Π i := Gi −

n+1

Πji := ∂˙j Π i ,

Gaa x˙ i ,

(11)

Πjki := ∂˙k Πji

for a given spray G in (U , h). It is easy to see that

Πjki = Gijk −

1 n+1

(δji Gaak + δki Gaaj + x˙ i Daajk )

(12)

and that a Πak = 0.

Dijkl := ∂˙l Gijk , called the (non-projective) Douglas tensor, transforms as a classical fourth-rank tensor. Its importance lies in the fact that Gijk are independent of x˙ l if and only if Dijkl = 0. That is, the vanishing of tensor D is necessary and sufficient for G to be a quadratic spray, as in classical affine geometry and its specialization to Riemannian geometry. If Gijk are constants in (U , h), then we say (9) is a constant spray and (U , h) is an adapted coordinate system. ¯ Π is called the normal spray connection in Furthermore, Πjki remains unchanged when G is projectively mapped onto G. (U , h) for G. Its spray curves are solutions of d2 x i ds2

¯

+ Πjki

dxj dxk ds¯ ds¯

= 0.

(13)

Remark. (1) s¯ remains unchanged under coordinate transformations (U , h) → (U¯ , h¯ ), whose Jacobians lie in SL(n, R), the real unimodular group on Rn , and only those (i.e. the structural group of T˜ M n is reduced from GL+ (n, R), the nonsingular real n × n matrices with positive determinant, to SL(n, R)). Since D = 0 for the production equations of the Margulis # 1

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model, normal coordinates exist at any point we care to consider in production space. This means that any geodesic through that point will have straight line equations in normal coordinates. Since these are usefully taken to be log biomasses, a weak form of Huxley’s Allometric Law does obtain. In fact, restricting to the unimodular group for the Jacobians of coordinate changes allows one to make simple translations within this local theory, so any experimental, data defined, log–log plot, can be translated to the origin (the chosen point). Thus, we arrive at a complete local version of HAL which holds for any quadratic production dynamics. (2) Πjki transforms as a classical connection (i.e. like Gijk above) if and only if transformations have constant Jacobian determinant. (3) Πjki is a tensor if and only if the structural group of T˜ M n is reduced to the transformation of coordinates (U , h) → (U¯ , h¯ ) of the form:

 x¯ i =

aij xj + bi

and

ck xk + h

b1



..  .   bn 

  ai  j  c1 · · · cn

h

(n + 1) × (n + 1) constant matrix with non-zero determinant. This is the classical projective group. Following the procedure of KCC theory in Appendix A, and performing path-deviation for spray Eq. (13), we obtain the analogue of the usual ‘‘geodesic’’ deviation equation: D 2 ui

+ Wji uj = 0,

ds¯2

(14)

where Wji = 2∂j Π i − ∂r Πji x˙ r + 2Πjri Π r − Πri Πjr .

(15)

This occurs as follows: We are given the local spray Π in (U , h) and let x (¯s; η) be a smooth 1-parameter family of solutions with initial conditions xi (0; η), x˙ i (0). Since a spray will have a solution through any point p ∈ U and in any direction, these are called arbitrary smooth initial conditions. By Taylor’s theorem, i

xi (¯s; η) = xi (¯s) + ηui (¯s) + η2 (· · ·) and substituting this into Πjki (x, x˙ ) passage to the limit η → 0, yields the variational equations d 2 ui

+ ∂l Πjki (x, x˙ )ul

dxj dxk

dxk duj

= 0. ds¯2 ds¯ ds¯ ds¯ ds¯ Defining the projective covariant differential operation as, for example, + 2Πjki (x, x˙ )

Ai/l := ∂l Ai + Πjli (x, x˙ )Aj ,

(16)

(17)

and DAi

:= Ai/l

dxl

,

ds¯ ds¯ with similar formulas holding for higher order tensors. Using this we can rewrite Eq. (16) as D ds¯



dui ds¯

+ Πri ur



+ Πli



dul ds¯

+ Πrl ur



  dΠri + 2∂r Π i − − Πli Πrl ur = 0, ds¯

(18)

(19)

which is precisely Eq. (14) because of Eq. (13) and the second degree homogeneity of Π i (x, x˙ ) in x˙ and Eq. (11). Now, following Berwarld’s technique, define 1

(∂˙k Wji − ∂˙j Wki ) 3 and (Weyl’s Projective Curvature) Wjki :=

i Wjkl := ∂˙l Wjki .

(20)

(21)

This 4-index quantity actually is a tensor. However, the projective covariant derivative of a tensor is not necessarily a tensor. We are now able to state the two main theorems of local projective differential geometry, [12]. i i Theorem A. There is a coordinate chart (U , h) on M n , n ≥ 3, such that Πjki = 0, if and only if, Wjkl = 0, and ∂˙l Πjki := Πjkl = 0. i The tensor, Πjkl , is called the (projective) Douglas tensor.

P.L. Antonelli et al. / Nonlinear Analysis: Real World Applications 12 (2011) 3238–3251

3243

i Theorem B. There is a coordinate chart (U , h) on M 2 such that Πjki = 0, if and only if, Πjkl = 0 and ρjkl = 0, where

ρjkl := rjk/l − rjl/k , rjk := Bhjkh , where

i s Bijkl = ∂l Πjki + Πjks Πsli + Πjls Πmk x˙ m − (k/l).

The symbol, −(k/l), means repeat all terms that come before but interchange k and l and put a minus in front of the whole expression. Remark. The four-index tensor B is analogous to the usual curvature of a spray except that Gij , Gijk are replaced by Πji , Πjki . The condition Πjki = 0 for all i, j, k ∈ {1, . . . , n} in some (U , h) coordinate system is the so-called condition of projective flatness. We now consider the n = 2 case of a normal spray connection curve (13) associated with a given constant spray. We know from Eq. (12) that 1 2 2 1 Π11 = −Π21 and Π22 = −Π12 . 1 2 1 2 ¯ We can therefore set Π11 = α¯ 1 , Π22 = β1 , Π22 = α¯ 2 and Π11 = β¯ 2 in Eq. (13), which becomes  1 2  2 2 d2 x 1 dx dx1 dx2 dx ¯1 + α ¯ − 2 β + α ¯ = 0, 1 2 2 ds¯ ds¯ ds¯ ds¯ ds¯  2 2  1 2 dx dx2 dx d2 x 2 1 ¯ ¯ ¯ + β1 − 2α¯ 1 dx ds + β2 = 0. ds¯2 ds¯ ds¯ ds¯

(22)

Now, 1 2 1 2 ρ121 = Π12 r11 + Π12 r21 − Π11 r12 − Π11 r22

(23)

from Theorem B. But, 1 2 2 1 r12 = Π11 Π22 − Π11 Π22 = r21 .

(24)

2 1 1 1 2 1 1 1 2 Π12 r21 − Π11 r12 = −Π11 [2Π11 Π22 − Π22 Π11 − Π22 Π11 ].

(25)

Also, Furthermore, 1 1 2 2 r22 = 2[Π22 Π22 − Π22 Π22 ],

(26)

1 1 2 2 r11 = 2[−Π11 Π11 + Π22 Π11 ],

so that substitution of Eqs. (24) and (26) into Eq. (23) yields

ρ121 = 0,

(27)

by using Eq. (25). Similarly, one can prove that

ρ212 = 0.

(28)

i It is now clear that ρjkl = 0. Also, Πjkl = 0 because in this constant connection case the normal spray is quadratic since, in general, i Πjkl = Dijkl − P



1 n+1

 δji Daakl −

1 n+1

yi ∂˙a Dajkl ,

where P means a sum of the three terms obtained by the cyclic permutation of j, k, l. Therefore, Πjki = 0 in some coordinate chart (U¯ , h¯ ). We have therefore proved the following theorem. Theorem C (Part I). Every two-dimensional constant spray is projectively flat, [13]. Remark. It is not true that there is a projective time-sequencing change from, say, d2 x 1 ds2 d2 x 2 ds2

= −2α2 = −2α1

dx1 dx2 ds ds dx1 dx2 ds ds

+ α1 + α2



dx2

2 −

ds



dx1 ds



2

2 

ds

 −

dx1

dx2

,

2 

(29)

ds

to d2 x1 /ds¯2 = 0, d2 x2 /ds¯2 = 0, by assuming that α1 , α2 are not zero. The reason is that Eq. (12) implies 1 Π22 ̸= 0,

2 Π11 ̸= 0,

¯ jki in Eq. (12), since Dijkl = 0 holds for Eq. (27). Theorem C states only that there is some coordinate system (U¯ , h¯ ) for which Π vanish. This is where the tensor character of Theorem A and B play an important role.

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Theorem C (Part II). In every dimension ≥ 3 there exists a constant spray which is not projectively flat. Proof. Consider the n-dimensional conformally flat Riemannian metric (gij ) = e2φ(x) .(δij ), with φ(x) = αi xi , αi constants. It is a well known fact that the Riemannian scalar curvature R is never constant and vanishes if and only if n = 2. Yet, the (geodesic) spray of this metric has constant coefficients. But, in Riemannian geometry, projective flatness is equivalent to constant sectional curvatures. Therefore, R must be a constant as well, and the proof is complete (see [11]).  Remark. There exist two-dimensional projectively flat Finsler metrics which are not of constant curvature [10,14]. Obviously, these cannot be Riemannian metrics. We have given the Margulis # 1 metric and geodesics equations above. Now, let φ be a smooth function of x1 and x2 and denote ∂i φ as φi and likewise φijk = ∂i ∂j ∂k φ , etc. Theorem D. For the Margulis # 1 Finsler metric, the geodesics are rectilinear for some coordinates x¯ i (so HAL holds) if and only iff ρ121 = 0, where

ρ121 =

λ(2λ + 1) [φ121 + λφ1 φ12 ] , 3(λ + 1)

and ρ212 = 0, where

ρ212

[ ] λ(λ − 1) λ = φ212 − φ2 φ12 . 3(λ + 1) λ+1

Remarks. (1) ρ121 = 0 ⇐⇒ λ = −1/2, since φ is arbitrary and also (2) ρ212 = 0 ⇐⇒ λ = 1, since, again, φ is an arbitrary smooth function of x1 , x2 . But, from remarks (1) and (2) above, we have a contradiction if both ρ121 = 0 and ρ212 = 0. We conclude that Margulis # 1 is not projectively flat, so cannot have rectilinear geodesics. We give now a different proof, one using Berwald’s formula IKs = −3Kb , characterizing projectively flat 2-dimensional Finsler spaces. (See Appendix A and [14] for definitions.)

φ = −α1 x1 + (λ + 1)α2 x2 + ν3 x1 x2 (y2 )1+1/λ F¯ = eφ . (y1 )1/λ 2G1 = −λφ1 (y1 )2 λ 2G2 = φ2 (y2 )2 λ+1 (λ + 2)2 I2 = λ+1 √  1+2/λ  λ + 1 y2 2φ . 1 g¯ = e . λ y l2 m1 = − √ g¯ l 1 m2 = √ g¯ li = ∂˙i F¯ . Notation:

δi K = ∂i K − Gri ∂˙r K G11 = −λφ1 y1

λ φ2 y2 λ+1 G12 = 0 = G21 li = yi /F¯  2 1+1/λ 1 y l1 = − eφ λ y1 G22 =

P.L. Antonelli et al. / Nonlinear Analysis: Real World Applications 12 (2011) 3238–3251

l2 =

λ+1 λ



 2 1/λ

y

y1

√

m1 = − λ + 1 m2 = − √



1

y1

eφ

1+1/λ

y2 e−φ

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

y1

e−φ

1/λ

y2 λ+1   1+2/λ y1 λ2 .ν3 . 2 .e−2φ K = λ+1 y

(δi K )mi = Kb δi K = ∂i K − Gri ∂˙r K ∂i K = −2φi K  1 2/λ ˙∂i K = λ(λ + 2) .ν3 .e−2φ . 1 . y λ+1 y2 y2  1 1+2/λ λ(λ + 2) y 1 ∂˙2 K = − .ν3 .e−2φ . 2 . 2 λ+1 y y δ1 K = λK φ1 λ δ2 K = − φ2 K λ+1  1 1/λ   √ y y1 φ2 −φ Kb = −λK e λ + 1φ1 2 − y2 y (λ + 1)3/2 Ks = (δi K )li Ks = λK e−φ



y1

1/λ 

y2

φ1

y1 y2

−

 φ2 . λ+1

Projective flatness holds iff IKs = −3Kb . Since φi are functions of x1 and x2 only, this identity implies both λ = 1 and λ = −1/2, a contradiction of K ̸= 0, φ1 , φ2 ̸= 0. This is an example of a 2-dimensional Berwald space with I 2 = 9/2 and with K ̸= 0 which is not projectively flat. Berwald provided many examples which are projectively flat [14]. On page 838 of the Handbook of Finsler Geometry, Matsumoto claims that a Berwald 2-space with I 2 = 9/2 and F = γ 2 /β , where γ and β are independent 1-forms, must be projectively flat (with rectilinear extremals). This is not Berwald’s theorem. It has been misunderstood by Prof. Matsumoto. Indeed, taking γ = eφ(x) .(y2 ), β = y1 , with φ(x) = −α1 x1 + (λ + 1)α2 x2 + ν3 x1 x2 , λ = 1, then I 2 = 9/2, from above, and noting F leads to a positive definite gij on a suitable positively conical region of T˙ M 2 , the slit tangent bundle, we exhibit a conter-example. Now, following Berwald, ds = F (x, dx) =

(dx1 + Z (x1 , x2 )dx2 )2 dx2

(30)

must satisfy d dγ

∫ γ

ds = 0.

Furthermore, from [10, Vol.II, page 801], we have

∂x1 ∂˙x2 F − ∂x2 ∂˙x1 F = 0

(31)

as a necessary and sufficient condition for Huxley’s Allometric Law (HAL). Using (30) and (31), one easily arrives at Z ∂x1 Z − ∂x2 Z = 0.

(32)

The solution Z = constant yields R = 0, but there are plenty of non-constant solutions Z (x, y). By integration we arrive at x1 + x2 Z = M (Z ),

(33)

where M (Z ) is an arbitrary smooth function of Z . Berwald also showed that R=

 3 y2 . . (M ′ (Z ) − x2 )3 y1 + Zy2 M ′′ (Z )

(34)

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Therefore, those F s for which M ′′ (Z ) > 0, M ′ (Z ) − x2 > 0 will be candidates for stable HAL production systems. The system equations are now written as

[ ] 1  2 1 2 dx1 dx d x 1 2 dx   = Φ (x , x ) ,  2 +ϵ dt dt dt dt [ ] 2  d2 x2 dx2 dx2  1 2 dx  + ϵ = Φ ( x , x ) . 2 dt

dt

dt

(35)

dt

where we have used the parameter transformation s = A − Bexp(−ϵ t ), ϵ > 0, the Douglas tensor and

φ(x1 , x2 ) =

1 M ′ (Z )

− x2

.

(36)

We know D = 0 for (35). From Theorem F in Appendix B we expect Rijkl or Rijk or K = R to be all non-vanishing. In the next section we prove that the expression (36) cannot be linear nor quadratic in x1 and x2 . 3. Theorem E Proposition 1. Zx = Φ (x, y) does not admit local extrema. We use the notation x = x1 , y = x2 , x˙ 1 = y1 and x˙ 2 = y2 . Proof. We know x + yZ = M (Z ), with M ′′ (Z ) ̸= 0. Clearly, yZxx = M ′′ (Z ).(Zx )2 + M ′ (Z ).Zxx

(37)

so that Zxx =

M ′′ (Z ).(Zx )2 y − M ′ (Z )

.

But, clearly, 1 + yZx = M ′ (Z )Zx , so Zx =

1 M ′ (Z ) − y

.

Therefore, Zxx = −M ′′ (Z ).(Zx )3

(38)

and, since any local extrema must satisfy

(Zx )x = 0 = (Zx )y we would have Zx = 0 at any such point (xo , yo ). This contradicts Zx = 1/(M ′ (Z ) − y) never zero.



Proposition 2. Zx = Φ (x, y) is not a linear nor a quadratic form in x, y. Proof. (A) Suppose Φ = ax + by + c. Fix any x = xo and solve to get yo = − ba xo − x = xo = −c /a. If a = b = 0, then Φ = c = Zx and (37) implies

c b

so that Φ (xo , yo ) = 0. If b = 0, take

M ′′ (Z ).c 2 = 0 so c = 0 (since M ′′ (Z ) ̸= 0). Therefore, in all cases of Φ linear we see that Zx = 0 at some point, which is impossible. (B) Suppose Φ = aij xi xj + bi xi + c (with x1 = x, x2 = y and summation convention). Clearly, Φx = 0 = Φy has always at least one solution (unique, if det(aij ) ̸= 0). Hence, Zxx = 0 and (38) imply Zx = 0 at this point.  Proposition 3. The Berwald I 2 = 9/2 geodesics are

[ ] 1  2 1 2 d x dx 1 2 dx   = Φ ( x , x ) ,  2 ds ds ds [ ] 2  d2 x2 dx2  1 2 dx  . = Φ ( x , x ) 2 ds

ds

ds

They are projectively flat (without any change of coordinates) and Φ (x1 , x2 ) is never linear nor quadratic. Proof. Proposition 2. From (38) we know Zxx = −M ′′ (Z )(Zx )3

(39)

P.L. Antonelli et al. / Nonlinear Analysis: Real World Applications 12 (2011) 3238–3251

3247

and since K = M (Z ).(Zx ) . ′′

3



3

y˙ x˙ + Z y˙

,

we have

 K = −Zxx

y˙

3

x˙ + Z y˙

.

Taking ∂x of K we get

3 [ ] 3Zxx Zx y˙ . −Zxxx + . x˙ + Z y˙ x˙ + Z y˙



y˙

Kx =

Therefore, Kx = 0 at (xo , yo ) = (x, y) ⇐⇒ Zxxx =

3Zxx Zx

(˙x/˙y + Z (x, y))

,

or, from (38), Zxxx = −

3M ′′ (Z )(Zx )4 x˙ /˙y + Z (x, y)

. 

Note: 1

x˙

.∂x˙ /˙y K H⇒ ∂x˙ K , ∂y˙ K never zero. (˙y)2 The left-hand side is just ∂x ∂x ∂x Z (x, y) and is independent of x˙ , y˙ or x˙ /˙y, since Z (x, y) is such, by definition. But, since M ′′ (Z ) ̸= 0 and Zx ̸= 0 everywhere in (x, y)-space, the derivative of the right-hand side with respect to x˙ /˙y is ∂x˙ K =

y˙

∂x˙ /˙y K ;

∂y˙ K = −

3M ′′ (Z )(Zx )4

(˙x/˙y + Z (x, y))2 and is never zero – a contradiction. Hence, Kx is never zero in (x, y)-space and (likewise, Ky ̸= 0∀(x, y)). We have proved Theorem E. K (x, y, x˙ , y˙ ) as a function on T˜ M 2 , the slit tangent bundle, cannot have local extremals. The only 2-dimensional Riemannian space with rectilinear extremals and K > 0 is the sphere, all points of which are local extremals since K = constant. For a next work, we intend to propose a combined model of Margulis #1 and Margulis # 2, in which to do so, we will interchange indexes # 1 and # 2 in Margulis # 1 and so the roles of parasite and host, and multiply the cost functional of Margulis # 2 by a conformal factor eφ(x1,x2) , φ(x1, x2) = −α2 x2 + (λ + 1)α1 x1 + ν3 x1 x2 . The idea here is that both models are contained in the combined one and if one starts with the interchanged # 1 model and allows the Z function, which starts out very small, to evolve to a greater influence, while letting the parameters of # 1 to have less and less influence, we end up with a system which satisfies Huxley’s Allometric Law and has stable production. Our intent is to investigate the quantitative biological aspects of this apparently non-heterochronic evolution. Appendix A Let (x1 , . . . , xn ) = (x), (dx1 /dt , . . . , dxn /dt ) = (dx/dt ) = (˙x), and t be 2n + 1 coordinates in an open connected subset Ω of the Euclidean (2n + 1)-dimensional space Rn × Rn × R1 . Suppose that we have d2 x i dt 2

+ g i (x, x˙ , t ) = 0,

i = 1 , . . . , n.

(40)

for which each g i is C oo in a neighbourhood of initial conditions ((x)0 , (˙x)0 , t0 ) ∈ Ω . The intrinsic geometric properties of (40) under non-singular transformations of the type



x¯ i = f i (x1 , . . . , xn ), t¯ = t ,

i = 1 , . . . , n,

(41)

are given by the five KCC-differential invariants, named after Kosambi [15], Cartan [16], and Chern [17], given below. Let us first define the KCC-covariant differential of a contravariant vector field ξ i (x) on Ω by

Dξ i dt

=

dξ i dt

1

+ g;ir ξ r , 2

(42)

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P.L. Antonelli et al. / Nonlinear Analysis: Real World Applications 12 (2011) 3238–3251

where the semi-colon indicates partial differentiation with respect to x˙ r , and we made use of the Einstein summation convention on repeated indices. Using (42), Eq. (40) becomes

D x˙ i dt

= ϵi =

1 2

g;ir x˙ r − g i ,

(43)

defining the first KCC-invariant of (40), the contravariant vector field on Ω , ϵ i , which represents an ‘external force’. Varying trajectories xi (t ) of (40) into nearby ones according to x¯ i (t ) = xi (t ) + ξ i (t )η,

(44)

where η denotes a parameter, with |η| small and ξ (t ) are the components of some contravariant vector defined along xi = xi (t ), we get, substituting (44) into (40) and taking the limit as η → 0, i

d2 ξ i

+ g;ir

dt 2

dξ r dt

+ g,ir ξ r = 0,

(45)

where the comma indicates partial differentiation with respect to xr . Using the KCC-covariant differentiation (42) we can re-express this as

D 2ξ i dt 2

= Pri ξ r ,

(46)

where

Pji = −g,ij −

1 2

g r g;ir ;j +

1 2

x˙ r g,ir ;j +

1 4

g;ir g;rj +

1 ∂ 2 ∂t

g;ij .

(47)

The tensor Pji is the second KCC-invariant of (40). The third, fourth and fifth invariants are:

 1 i i i   Rjk = (Pj;k − Pk;j ), 3

(48)

i B i = Rjk  ;l ,   jkl i i Djkl = g;j;k;l .

The main result of KCC-theory is the following: Two systems of the form (40) on Ω are equivalent relative to (41) if and only if the five KCC-invariants are equivalent. In particular, there exist coordinates (¯x) for which the g i (¯x, x˙¯ , t ) all vanish if and only if all KCC-invariants are zero. The tensor D vanishes if and only if g i is quadratic in x˙ , in the case when the first KCC-invariant vanishes. Let us now introduce the notion of an n-dimensional Finsler space as a manifold where, given a coordinate system (x) and a curve xi = xi (t ), the norm of a tangent vector x˙ i to the curve at each point P on xi (t ) is given by the positive metric function F , |˙xi | = F (x, x˙ ), where F is positively homogeneous of degree 1 in x˙ i . From F , a metric tensor is defined as gij (x, x˙ ) = (∂ 2 F 2 /∂ x˙ i ∂ x˙ j )/2, which must be regular in an open region of the tangent bundle, the collection of all tangent vectors to the manifold, and which excludes the origins. The use of the calculus of variations for F leads to (40) with g i (x, x˙ , t ) = γjki (x, x˙ ) x˙ j x˙ k , where the γjki are the Levi–Cività symbols for the Finsler metric tensor gij (x, x˙ ). Berwald’s Gaussian curvature K for 2-dimensional Finsler spaces is defined from his famous formula [11] i Rjk = F K mi (lj mk − lk mj ),

where

i Rjk

(49)

is given by the first equation in (48), l = x˙ /F is the unit vector in the x˙ direction, and m the unique (up i

i

i

i

to orientation) unit vector perpendicular to li .3 Lowering the index on mi via the metric tensor gives mi , which satisfies F (x, m) = gij (x, x˙ ) mi mj ≡ mi mi = 1. If our curvature K is bigger than zero everywhere, then trajectories oscillate back and forth, crossing the reference trajectory. In this case, we say (40) is Jacobi stable. If K ≤ 0 everywhere, trajectories diverge and system (40) is Jacobi unstable [10,18]. This notion of stability is a Lyapunov notion, but it is a whole trajectory concept. The famous torsion tensor of Cartan is defined after (58). Using the Berwald frame (li , mi ) for n = 2, it can be proved that FCijk = Imi mj mk for a scalar I, which is unique up to orientation, therefore I 2 is uniquely defined. Appendix B Our standard reference here is [10, Vol. I, Part 2]. All manifolds will be C ∞ without boundary. Let M n be an n-dimensional manifold. By parallel transport on M n we mean the existence of linear (Kozul) connection. A Finsler connection is a linear

3 The 3-index R in (49) is in the case of a spray, identical to the 3-index R in Appendix B, (63). Likewise, K is identical with R = K in Appendix B.

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(Kozul) connection D on TM n , the tangent bundle on M n with zero section deleted, which preserves under the action of D the Whitney sum decomposition TTM = HIM ⊕ VTM of horizontal and vertical distributions. Thus, DH = 0 = DV . The HTM subbundle of the double tangent bundle of M is often called a nonlinear connection on M n . We define the covariant derivative (induced by HTM) of a vector field (C ∞ ) on M n with local components X i by

∇ X i = ∂j X i yj + Nji X j

(50)

where (x , . . . , x , y , . . . , y ) are local coordinates on TM. The summation convention on repeated upper and lower indices i is used throughout. Also, ∂j ≡ ∂∂xj throughout this paper. The n2 functions Nji transform under (xi ) → (x ) non-singular just 1

n

1

n

as γjki (x)yk , where γ is the classical Levi–Cività connection of Riemannian geometry. The n2 -quantities are called the local coefficients of the non-linear connection. Define now the Berwald basis for the set of all vector fields on TM n by

{δi , ∂˙i },

∂ δi = ∂i − Nij (x, y)∂˙j , ∂˙j = j . ∂y

(51)

Using this basis we can define the local coefficients of the Finsler connection D as follows:

Dδi δj = Fjik (x, y)δk ,

Dδi ∂˙j = Fjik (x, y)∂˙k

D∂˙i δj = Cjik (x, y)δk ,

D∂˙i ∂˙j = Cjik (x, y)∂˙k .

(52)

Under non-singular coordinate change (xi ) → (x ) the Fjik (x, y) transform just as a classical linear connection (like, say γjki (x)) i

while

Cjik

(x, y) is a tensor.

Denote {δi , ∂˙i } by {Xa }a=1,2n and by {θ a }a=1,2n , the dual basis {dxi , δ yi }. The connection 1-forms (ωba ) corresponding to θ a are defined as

ωji = Fjki dxk + Cjki δ yk

(53)

and 1st-structure equations for D are



−dxh ∧ ωji = −Θ i i d(δ yi ) − δ yh ∧ ωhi = −Θ

(54)

 i } are given as where the 2-forms of torsion Θ a = {Θ i , Θ Θi =

i = Θ

1 2 1

Tjki dxj ∧ dxk + Cjki dxj ∧ δ yk Rijk dxj ∧ dxk + Pjki dxj ∧ δ yk +

2 The 2-forms of curvature for D are

1 2

(55) Sjki δ yj ∧ δ yk .

dωji − ωjh ∧ ωhi = −Ωji

(56)

where 1 i 1 i i Rjkh dxk ∧ dxh + Pjkh dxk ∧ δ yh + Sjkh δ yk ∧ δ yh . (57) 2 2 For a Finsler metric on M, we shall need to use the so-called Cartan–Finsler connection, but there are a number of other very important connections, [10]. For Cartan we must require

Ωji =

Tijk = 0,

Sijk = 0,

Pijk = 0.

(58) L2

It will follow that Cijk = gil Cjkℓ =

1 ˙ ˙ ˙ 2 ∂ ∂ ∂ L , where L(x, y) is the so-called Finsler metric function. If we set  F = 2 , then 4 i j k  ˙ ˙ gij (x, y) := ∂i ∂j F is the so-called fundamental metric tensor g = (gij ) of the Finsler manifold (M n ,  F ). However, the Levi–Cività coefficients γjki depend on yi , as well as xi and are not connection coefficients in Finsler geometry proper (i.e. Cjki ̸= 0 ⇐⇒ gij

depends on yi ). But, the local coefficients of Cartan satisfy Fjki (x, y) =

1

g ir (x, y) δk gjr (x, y) + δj gir (x, y) − δr gjk (x, y) ,





(59) 2 which is, in fact, very similar in form to the famous Levi–Cività formula which, by replacing δi by ∂i , gives precisely that formula. The last two basic properties of the Cartan connection for (M n ,  F ) are horizontal and vertical metricity: if we denote this most important connection by C Γ = (Γ h , Γ v ) where Γ h is given by Fjki (x, y) and Γ v by Cjki (x, y), then globally, h-metrical : ∇ h g = 0,

v -metrical : ∇ h g = 0,

(60)

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P.L. Antonelli et al. / Nonlinear Analysis: Real World Applications 12 (2011) 3238–3251

where the first is given locally as

δk gij = Fijk + Fjik ,

(61)

ℓ

where Fijk = giℓ Fjk , and the second is given locally by

∂˙k gij = Cijk + Cjik = 2Cijk .

(62)

Definition. A vector field S on TM is a semispray if and only if S = yi ∂∂xi − 2Gi (x, y)∂˙i locally on TM. Gi are the local coefficients and HTM is the induced nonlinear connection, Nji = ∂˙j Gi = Gij . the Berwald connection D induced by HTM. This is a Finsler connection with the local coefficients BΓ =  Consider  ∂ 2 Gi ∂ Gi i i i Nj = ∂ yj , Fjk = ∂ yj ∂ yk , Cjk = 0 . The connection 1-forms of the Berwald connection D are then given by

ωji = Fjki dxk =

∂ 2 Gi dxk = Gijk dxk . ∂ yj ∂ yk

The Berwald connection has only one component of torsion, the v(h)-torsion, which also gives the 3-index curvature of the nonlinear connection:

vT



δ δ , δ xi δ xj



 =

∂˙ =

Rkji k

δ Njk

δ Nik − δ xi δ xj

 ∂˙k .

(63)

The horizontal 2-forms of torsion Θ i of the Berwald connection vanish and the vertical 2-forms of torsion of the Berwald connection are given by: 1 ˜ i = Rijk dxj ∧ dxk . Θ 2 The two nonzero components of curvature for the Berwald connection D are:

Dh ijk

δ Fhji

 i δ Fhk m i m i   + F F − F F ; hj mk hk mj  δ xk δ xj  ∂ 3 Gi   = h j k. ∂y ∂y ∂y

Rh ijk =

−

(64)

The curvature 2-forms of the Berwald connection are given by: 1 Ωji = Rj ikh dxk ∧ dxh + Dj ikh dxk ∧ δ yh . 2 The first structure equations of the Berwald connection D are given by:

−dxh ∧ ωhi = 0, (65)

1 d(δ yi ) − δ yh ∧ ωhi = − Rijk dxj ∧ dxk . 2 The second structure equations of the Berwald connection D are given by: 1 dωji − ωjh ∧ ωhi = − Rj ikh dxk ∧ dxh − Dj ikh dxk ∧ δ yh . 2

(66)

Theorem F. The Berwald connection of a semispray S has zero curvature (is flat, i.e. R = 0, D = 0 in (64)) if and only if about every point p ∈ M there are local coordinates (xi ) in M such that with respect to the induced coordinates (xi , yi ) on TM, the local coefficients of the semispray S have the form: 2Gi (x, y) = Aij (x)yj + Bi (x).

(67)

Proof. If there exist induced coordinates on TM such that the semispray S has the local coefficients 2Gi (x, y) = Aij (x)yj +Bi (x), then the local coefficients of the Berwald connection D vanish, that is Fjki = ∂∂yj ∂Gyk = 0. From (64) we can see that the curvature components of D vanish so the Berwald connection is flat. Now let us assume that the curvature 2-forms Ωji of the Berwald connection vanish. As the horizontal torsion 2-forms Θ i are zero, there are induced coordinates on TM with respect to which the local coefficients of the Berwald connection vanish: 2 i

Fjki = 0 and Cjki = 0. But Fjki = ∂∂yj ∂Gyk , so with respect to these coordinates we have that 2Gi (x, y) = Aij (x)yj + Bi (x). 2 i



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