Evolutionary Dynamics and Lipschitz Maps
Abstract
In CLEVACKTHI (); CLEVACK () an attempt is made to find a comprehensive mathematical framework in which to investigate the problems of wellposedness, asymptotic analysis and parameter estimation for fully nonlinear evolutionary game models. A theory is developed as a dynamical system on the state space of finite signed Borel measures under the weak star topology. Two drawbacks of the previous theory is that the techniques and machinery involved in establishing the results are awkward and have not shed light on the parameter estimation question. For example, in CLEVACK () the proof for the existence of the dynamical system is obtained via a fixed point argument using the total variation topology, however, the continuity of the model is established in the topology. This has caused some confusion. I have remedied this by making all the vital rates Lipschitz and the dynamical system is defined on the dual of the bounded Lipschitz maps, a Banach space. I introduce a method of multiplying a functional by a family of functionals. This multiplication behaves nicely with respect to taking normed estimates. It allows us to form a semiflow that is locally Lipschitz, positive invariant, and covers all cases: discrete, continuous, pure selection, selection mutation and measure valued models. Under biologically motivated assumptions the model is uniformly eventually bounded. This remedies both the above problems as only one norm is used, this norm induces the topology on the positive cone of measures and since we have a norm and local Lipschitzity we can form a theory of Parameter Estimation.
keywords:
Evolutionary game models, selectionmutation, measure valued model, continuous dependence, darwinian evolution, Lipschitz mapsMsc:
[2010] 91A22, 34G20, 37C25, 92D25.1 Introduction
Evolutionary games (EG)s are a great unifying tool of population dynamics. Models ranging from a basic homogeneous parameter logistic model, to a parametric heterogeneous juvenile adult or consumer resource population model can be modeled effectively AFT (); JC3 (); JC2 (). For these type models, effective modeling means that one can study well posedness, asymptotic analysis and parameter estimation in one abstract setting. An initial attempt was made in CLEVACK (), however, this involved using several different topologies to establish the dynamical system and it shed no light on the parameter estimation question. We remedy the first problem in this manuscript and provide an introduction to a remedy for the second. The remedy consists of formulating a dynamical system on the dual of the bounded Lipschitz maps and making all of the vital rates Lipschitz continuous mappings.
As a brief recap, we mention again the reasons for the development of this abstract machinery. We consider the following EG (evolutionary game) model of generalized logistic growth with pure selection (i.e., strategies replicate themselves exactly and no mutation occurs) which was developed and analyzed in AMFH ():
(1) 
where is the total population, is compact and the state space is the set of continuous real valued functions . Each is a two tuple where is an intrinsic replication rate and is an intrinsic mortality rate. The solution to this model converges to a Dirac mass centered at the fittest class. This is the class with the highest birth to death ratio , and this convergence is in a topology called (point wise convergence of functions) AMFH (). However, this Dirac limit is not in the state space as it is not a continuous function. It is a measure. Thus, under this formulation one cannot treat this Dirac mass as an equilibrium (a constant) solution and hence the study of linear stability analysis is not possible.
Other examples for models developed on classical state spaces such as that demonstrate the emergence of Dirac measures in the asymptotic limit from smooth initial densities are given in AFT (); AMFH (); calsina (); CALCAD (); GVA (); P (); GR1 (); GR2 (). In particular, how the measures arise naturally in a biological and adaptive dynamics environment is illustrated quite well in (P, , chpt.2). These examples show that the chosen state space for formulating such selectionmutation models must contain densities and Dirac masses and the topology used must contain the ability to demonstrate convergence of densities to Dirac masses. This process is illustrated in the precursors to this work in CLEVACKTHI (); CLEVACK (). However in this manuscript I shall concentrate on the problems mentioned in the first paragraph.
Definition 1.1.
If is a metric space, and is an interval that contains zero then a map
is called a local (global autonomous) semiflow if:


,
If is a locally Lipschitz vectorfield and is the unique solution to and . Then we obtain a global autonomous semiflow This semiflow is always continuous (Thi03, , Chpt.1, pg.19).
In particular, in the present paper we let be our metric space where
Here is a compact metric space and are the bounded Lipschitz maps on is the dual of and are the Lipschitz maps into . Elements of are to be thought of as generalizations of probability measures. They are elements of of norm 1. is the parameter of our system and is to be thought of as a family of “probability distributions” indexed by . It is the mutation kernel. The metric satisfies
(See subsection 3.2 for the definitions of and )
In order for a semiflow to model our Evolutionary Game it must satisfy the constraint equations. In other words our (EG) model is an ordered triple
subject to:
(2) 
Here is the strategy (compact metric) space, is a semiflow on and
is a vector field (parameter dependent) such that and satisfy equation (2).
Our original models in this field all showed convergence of a semiflow generated from an initial value problem. The equilibrium point was a dirac mass. The obvious choice for state space was , under the topology. Where denotes the cone of the positive measures. However, is a complete metric space and not a Banach Space. With slight modifications of the definitions one could use the techniques of either mutational analysis JPA1 (); JPA2 (); Lorenz () or differential equations in metric spaces Tabor02 () or arcflows of arcfields CC (); CB () to generate a semiflow that satisfies the equivalent of the initial value problem in semiflow theory language.
The method employed here is that we find a Banach Space, containing as a closed metric subspace. Then we extend the constraint equation on to one on . The semiflow resulting from the solution of the generalized constraint equation has as a forward invariant subset and hence we generate our semiflow on . This is essentially the method employed here. However, using this approach we see that we generate a semiflow on any forward invariant subset of .
The main contributions of the present work are as follows:

We form a well posed model of a general evolutionary game as a semiflow on a suitable metric space that covers discrete, continuous, pure selection, selection mutation, and measure valued models. It should be noted that the pure selection kernel is Lipschitz and Lipschitz continuous function are dense in the continuous ones, since is compact.

Unlike the linear mutation term commonly used in the literature, we allow for nonlinear (density dependent) mutation term that contains all classical nonlinearities, e.g., Ricker, BevertonHolt, Logistic;

Unlike the one or two dimensional strategy spaces used in the literature, we allow for a strategy space that is possibly infinite dimensional. In particular, we assume that is a compact metric space.

Our state space has a norm and hence all estimates in the state space are performed with one metric which is a norm. This fact allows us to construct a theory of parameter estimation. This latter remedies the problems with the previous approaches.
This paper is organized as follows. In section 2 we demonstrate how to proceed from a density model to a valued model and thus demonstrate the derivation of the constraint equation. In section 3 we establish some background material including notation and technical definitions. In section 4 we prove positive invariance and well posedness. In section 5 we mention and demonstrate the unifying power of this methodology and mention that with the new formulation we still have both pure selection and selection mutation formulated in a continuous manner since the pure selection kernel is Lipschitz. In section 6 under a biologically motivated assumption we show uniform eventual boundedness. In section 7 we provide concluding remarks.
2 The Constraint Equation
This abbreviated section is taken from CLEVACK () just for background and the defining of the constraint equation. For the full account, see CLEVACK (). We begin with a density version of the constraint equation (3). To this end take as the strategy space a compact subset of (the interior of the positive cone of ) and consider the following density IVP (initial value problem) :
(3) 
Here, is the total population, represents the densitydependent replication rate per individual, while represents the densitydependent mortality rate per individual. The probability density function is the selectionmutation kernel. That is, represents the probability that an individual of type replicates an individual of type or the proportion of ’s offspring that belong to the ball. Hence, is the offspring of in the ball and is the total replication of the ball into the ball. Summing (integrating) over all balls results in the replication term. Clearly represents the mortality in the ball. The difference between birth and death in the ball gives the net rate of change of the individuals in the ball, i.e., Dividing by we get (3).
We point out that formally, if we let (the delta function is even) in (3) then we obtain the following pure selection (density) model
(4) 
of which equation (1) in AFT () is a special case. Indeed if then this means that the proportion of ’s offspring in the ball is zero unless in which case this proportion is i.e., individuals of type only give birth to individuals of type .
Multiplying both sides of (3) by a test function and integrating over we obtain:
Changing order of integration we get
where See CLEVACK () for a more biological interpretation of the mutation kernel.
If we obtain the following measure valued dynamical system
(5) 
if
If we properly define the operation below, then we obtain the following valued model:
(6) 
if
or for
(7) 
(7) is the valued constraint equation.
3 Preliminary Material
We begin modeling with where is a compact metric space, are the Borel sets on and is a probability measure on the Measurable Space representing an initial weighting on the strategies. One can think of as a compact subset of and as a probability measure (initial weighting) on this set. above is used to model the space of strategies. What we seek as a model of our game is a semiflow subject to the constraint equation (7) which will follow easily from a parameter indexed family of solutions to (5) above.
3.1 Birth and Mortality Rates
Concerning the birth and mortality densities and we make assumptions similar to those used in AFT ():

is locally Lipschitz continuous and nonincreasing on for any

is locally Lipschitz continuous, is nondecreasing on for any , and . (This means that there is some inherent mortality not density related)
These assumptions are of sufficient generality to capture many nonlinearities of classical population dynamics including Ricker, BevertonHolt, and Logistic (e.g., see AFT ()).
3.2 Technical Preliminaries for Bounded Lipschitz Formulation
If is a Banach Algebra, denotes the continuous  valued maps under the uniform norm,
Two important subspaces are
Where is the dense subspace of all valued Lipschitz maps and is the locally compact subspace of Lipschitz maps with Lipschitz bound smaller than or equal to .
If no range space is specified then , denotes the Banach space of continuous real valued functions on . The two important subspaces mentioned above are denoted as and respectively.
also has a finer structure. Indeed, if define
Under the norm , becomes a Banach space denoted
denotes the continuous dual of this Banach Space and it has a closed convex subspace
(8) 
and are the same set, the topology is just different.
Crucial to the success of our modeling efforts is the forming of the parameter space, which models the mutation kernel. It is a convex subset of
Some Algebra :
Firstly we note that both and are also Banach Algebras and we have the inequality
(9) 
holding in each space.
Secondly, we view as a family of bounded linear functionals indexed by . It has properties that need elucidating for our modeling purposes. is a unital BL module. Indeed if
(10) 
We will denote this action simply as since it is just pointwise multiplication. So one can multiply a family of functionals by a Lipschitz map and obtain another family of functionals. Moreover, the new uniform normed product is no larger than the uniformed product of the norms.
Thirdly,
is an isometry. Where is the delta functional.
This allows us to view a Lipschitz function, , as a family of bounded linear functionals on indexed by Moreover this viewing preserves the uniform norm, i.e.
Fourthly, we need to “multiply” a functional by a family of functionals. Let denote the normed Algebra of bounded maps of into where we have pointwise addition and multiplication and the norm defined as
If
then is a  Algebra under pointwise addition and multiplication and is a  module. Indeed, under the action
given by
we have an action. This is a bounded Lipschitz functional since is bounded and Lipschitz since With respect to the normed product we have
(11) 
Moreover, if , (11) becomes
(12) 
where
above allows us to “multiply” a functional, , by a family of functionals
This new multiplication gives us some important information about our mutation parameter space
Indeed,

First notice
If we think of as (same set different topology), then we actually have that
by
(13) The operation does not make into a  module since is not a ring .^{3}^{3}3 Since acts as a sort of identity. It is more than likely that some sort of convolution product could be placed on with being the identity. Then it would be a unital module. However, this restriction of is bilinear.

Also note that if , then is well defined as well. Indeed, from the thirdly observation in the Some Algebra section we view as the family , and
(14) Furthermore
(15) 
If , and , then is possibly in For example, suppose that then But even though , is possibly an element of For instance, if and for some

In all cases behaves nicely with respect to norm estimation in all norms. The normed product is no larger than the product of the norms.
Miscellaneous:
If ,
Below and likewise for
0 denote the zero functional and denotes the constant function that takes the value .
For any time dependent mapping, , we let
4 Main WellPosedness Theorem
The following is the main theorem of this section.
Theorem 4.1.
Let Then is a metric space where
Moreover there exists a global autonomous semiflow where
satisfying the following:

There exists a continuous mapping
such that

For fixed , the mapping is the unique solution to
(16) Moreover, if
by
Then
(17) 
If , then is forward invariant under i.e. .

if then is Lipschitz continuous on
We now establish a few results that are needed to prove Theorem 4.1.
4.1 Local Existence and Uniqueness of Dynamical System
With this background we prepare to obtain the semiflow that will model our evolutionary game. If is the vectorfield defined in (16) then
(18) 
and
(19) 
For each , define as follows. If is one of the functions then we extend to by setting for for . Then is bounded and Lipschitz continuous. Let be the redefined vector field obtained by replacing with
For each , we will resolve the following IVP first.
(20) 
where
(21) 
and
(22) 
Lemma 4.2.
(Lipschitz )

, there exists continuous

, if
or
then is bounded and Lipschitz.
Proof.
First notice that follows from since
(23) 
and
We will prove the second condition in . The first is straightforward and the only real difference in the argument used below is that one uses the estimate in 11 instead of the estimate in 12. If , , , then let be as in 21. Then
(24) 
Hence,
(25) 
If
then
Since is bounded and Lipschitz on ∎
Lemma 4.3.
(Estimates) Let If and we have the following estimates:


As a function of ,
(26) 
If
(27)

Proof.

Using the mean value theorem on the function, , there exists such that

Using the mean value theorem on the function, , there exists , such that
∎
Proposition 4.4.

For each , , is the unique solution to
(28) in


is Lipschitz continuous on
Proof.
For and , define
It is an exercise to show that is a Banach space. In fact and
are equivalent.
Unique local solution to (28):
Using standard techniques for locally Lipschitz vector fields with a parameter into a Banach space, Lemma 4.2 relays that we have a unique solution to (28) on for any We can use a Lipschitz argument similar to the one below to show that this mapping is indeed Lipschitz.
We label this solution (to denote the dependence on ).
Forward invariance of :
Let if is as above define
Obviously is a nonempty closed subspace of and hence is a complete metric space.
If define