The population dynamics of vertically and horizontally transmitted parasites

شرح پروژه

For each project the tasks are the same,namely,

1. The general k\[Dash]morphic resident
>Give the equations for the dynamics of a k\[Dash]morphic resident popoulation
of strategies Subscript[x, 1],\[Ellipsis],Subscript[x, k] and corresponding populatioin densities Subscript[n, 1],\[Ellipsis],Subscript[n, k];
>Specify the invasion fitness of a mutant strategy y in a k\[Dash]morphic resident 
population of strategies Subscript[x, 1],\[Ellipsis],Subscript[x, k]; What can you say about the effective 
dimension of the feedback environment? What does that imply for the maximum 
number of resident types that could possibly coexists? ;

 2. The virgin environment
>What does the virgin environment (k=0)look like? ;
>Give the invasion fitness  of strategy y in the virgin environment? Which 
strategies can invade? ;

3.The monomorphic resident 
>Give equations for the monomorphic resident (k=1),and study its dynamics;
In particular,find all positive attractors and specify how they depend on
the resident strategies; This may be done analytically,(if possible) or 
numerically;
>Give an expression for the invasion fitness of a  mutant strategy y in a
resident population with strategy x; N.B.,if there are multiple positive
resident attractors,then the invasion fitness has to be specified for each
attractor separately;
>Make a PIP for given parameter values; If there are multiple positive
resident attractors,then each will require its own PIP; Are there any singular
strategies and if so,what type are they?; Interpret the PIP in terms of  
strategy dynamics;
>Make a bifurcation plot of the singular strategies as a function of the 
parameter values (only one parameter at a time), and indicate which 
singularities are repelling/attracting and invadable/uninvadable; 

4. The dimorphic resident
>Give equations for the dimorphic resident ( k=2)and study its dynamics;
In particular,find all positive attractors and specify how they depend on the
resident strategies,but only if you need those for the calculation of the 
invasion fitness; This may be done analytically,(if possible) or numerically; 
If the essential dimension of the environment is equal to 2, then there is no
need to study the resident dynamics (see demo project);
>Make a mutual invadability plot to see which strategies can coexist;
>Give an expression of the invasion fitness of a rare mutant strategy y in a
dimorphic resident population with strategies Subscript[x, 1] and Subscript[x, 2]; 
>Make isocline plots for given parameter values,i.e.,indicate where in
the coexistence region the selection gradient of one or the other resident 
strategy becomes  zero; Interpret the isocline plot in terms of the strategy
dynamics;

5. Singular function analysis (BONUS)
> If applicable to your project,you may want to do a critical function analysis
(see project descriptions);
 

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