The basic equation in ecology, evolutionary dynamics, and several other studies is the logistics curve or growth curve with limits. 

                                                          (10.1)

Where    = the size of the population,

 = the rate of increase of , and

= the maximum size of .

This simple growth equation can be elaborated to include the cooperative, competitive, and antagonistic growth between two different populations.  Here cooperation means that an increase in the population of one entity type directly increases the rate of growth of another type.  Competition means that two entity types require a common resource.  Antagonism means that an increase in the population of one entity type directly reduces the rate of growth of another type.

Cooperation, 
                                                                                                                  (10.2)

Competition,
                                                                                                                      (10.3)

Antagonism,
                                                                                                                  (10.4)

 The last two can be combined to yield

 Competitive Antagonism, 
                                                                                                           (10.5)

The reader, familiar with ecological studies, will immediately recognize an elaboration here.  In the biological realm, competition is considered to be a contest for some common resource.  But, there is another form of direct competition, which I have labeled antagonism.  This occurs in ecology when two species prey on each other.

Figure 10.1: Meme Growth Examples

Simple Growth,              ®

Cooperation,                  +®, à®

Antagonism,                  D®, ´®

, , , ,  

Figure 10.1 plots equations 10.1, 10.2, and 10.4, i.e.  simple growth, cooperation, and antagonism.  Here we see that cooperation between two memes increases the rate of growth of both.  One meme has been given a slightly greater growth rate to give a feel for the sensitivity to this parameter.  The graph also depicts the effect of antagonism between memes.  Notice the rather dramatic effect of a very small difference between the rates of growth between antagonistic memes.

Figure 10.2 shows the growth curve for competitive, and competitive antagonism, with the simple growth curve for comparison.  The clear effect is to prevent competing memes from spreading to the entire population.

Figure 10.2: Meme Growth Examples

                        Simple Growth,                          ®

                        Competition,                              +®, à®

                        Competitive Antagonism,            D®, ´®

                        , , , ,

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(C) 2005-2014 Wayne M. Angel.  All rights reserved.