The model
An individual-based model (IBM) was designed to investigate the
influence of some populational parameters on the colonization success of
a new host species. During simulations, pathogens with variable
propagule sizes, reproduction rates, and rates of emergence of
phenotypic novelties were challenged by new host species representing
different levels of compatibility (which are related to the selection
pressure that the new host represents). The consumer-resource system can
be applied to several different types of symbioses and ecological
associations; for simplicity, hereafter we will designate these as the
host-pathogen interaction. The model (written in Fortran) is available
through Github
(https://github.com/sofiagalvao2020/SimpleHost_switching).
Pathogen and host descriptions:
Each pathogen i is described by a compound phenotype
(=fundamental capacity space as defined in Agosta and
Brooks 2020) of G binary individual phenotypes. The binary phenotypes
can assume the values of either one or zero, which can be understood as
the expression of two distinct traits within the same locus or set of
loci. The sum of all characters defines the individual’s compound
phenotype (=realized capacity space of Agosta and Brooks 2020),
which can vary between 0 and G. This capacity space is composed
by inheritable features, subjected to change over generations, and under
selection according to its compatibility to the host. The compound
phenotype is labeled as pi,n , in which the
subscripts identify the pathogen i of the generation n .
For the beginning of the simulation, the sum of all “loci” is identical
for all propagule individuals- creating a standard populational compound
phenotype p0 at the start of the colonization
attempt.
For simplicity, as in Araujo et al. (2015), the host is characterized by
a single number (ph ) which represents the optimum
value of the compound phenotype imposed on pathogens. It is a fixed
throughout the simulation. Here we assume ph =
G/2. Besides defining an interaction pressure around this optimum value,
the host is also represented by a carrying capacity on the pathogen
population of K individuals.
Dynamics
The dynamics starts with a propagule size of N0pathogen individuals challenged to colonize the host - there is only one
colonization attempt per simulation. For simplicity, their phenotypes
are randomly defined, but when more than one pathogen individual is
considered, they present identical compound phenotypespi,n=0 =p0 ∀ i –
that is, they have the same fitness in the new host but carry different
phenotypes (i.e. represent distinct fundamental capacity space ).
Each iteration step represents a generation n where the pathogens
will undergo Selection and Reproduction (Fig 1), as
detailed below.
Selection
The selection is imposed as the survival probability of each pathogeni in a given generation n and it follows a normal
distribution:
\(P_{\text{survival}}=exp\left[\frac{-{{d{}^{2}}_{i,n}}}{2}\right]\), (1)
where
\(d_{i,n}=\frac{p_{i,n}-\text{\ p}_{h}}{\sigma}\) (2)
is the distance between the pathogen compound phenotype (pi,n ) and the optimum imposed by the host (ph ) in units of the deviation rate (σ). The
deviation rate represents the selection strength imposed by the new host
- the larger the deviation rate, the larger is the diversity of
phenotypes that are capable of surviving on that specific host (Fig 1).
For the propagule population - with all individuals presenting the same
compound phenotype p0 - the initial phenotype
distance from the propagule to the host isd0= (p0 -ph )/
σ. The model imposes this survival probability (Eq. 1) to every
individual, and the survivors (Ns,n ) go to the
next model step, Reproduction .
Reproduction
At this step, the pathogens that survived the previous step
(Ns,n ) produce offspring depending on the reproduction rate
(b, the average number of descendants per parental) and the
carrying capacity (K ). For simplicity, we assume asexual
reproduction. The number of descendants for the next generationn+1 will be Ns,n*b if this value does not
exceed K , otherwise, the number of descendants is K .
Random individuals of the surviving population are selected to generate
one offspring with reposition - the progenitor can be selected more than
once. This process is repeated until the total number of descendants is
achieved. Each descendant inherits the same chain of characters of its
progenitor with a probability μ of incorporating a novelty per
locus (i.e. changing from 0 to 1 or from 1 to 0). After all reproduction
events, the progenitors die and the descendants constitute the next
generation that will be subjected to the new Selection andReproduction cycle (Fig 1).
The rate novelty emergence (μ) refers to any kind of novelty
introduced into the pool of capacity of the individual, indirectly
influencing the pathogen’s fitness to the host. These evolutionary
novelties can emerge, accumulate, and be maintained throughout
generations simulating inheritance mechanisms, comprising thecapacity space of the pathogen (called information spacein Brooks and Agosta 2012, Jablonka et al. 2014, Brooks et al. 2019; see
also Agosta and Brooks 2020). We refrain from using “mutation rate” -
as opposed to “rate of emergence of evolutionary novelty” - to avoid
the strictly genetic meaning of the expression used in the Modern
Synthesis (see Brooks and Agosta 2012; Laland et al. 2015; Agosta and
Brooks 2020).
Simulations and data analyses
For each parameter combination, we ran 700 simulation repetitions for
1,000 generations or until the pathogen population went extinct. We then
calculated the proportion of simulations without extinction and defined
it as the probability of successful establishment . The
sensibility of the probability of successful establishment to each
parameter was calculated by varying two of them and fixing the remaining
ones (the fixed values are highlighted in Table 1). The parameterp0 was always one of the varied parameters and it
varied between the fittest (p0=ph=G/2 ) to the least fit value
(p0 =G ). Given that the propagule survival
probability (Eq. 1) depends only on d0 , we fixed\(\sigma=10\ \)and, as a consequence, the propagule compound
phenotype distance from the host varied according to\(0{\leq d}_{0}\leq G/20\). The investigated values of novelty rate
(μ ) are 0, 10-6,10-5,
10-4, 10-3, 10-2,and 10-1. Higher novelty values, such as
10-2 and 10-1 are considered analogs
of the high mutation rates observed in viruses (e.g. Drake and Holland
1999). Furthermore, although biologically unreal, the null value forμ represents the inferior limit of our analysis. We also varied
the reproduction rate (b ), propagule size
(N0 ,), compound phenotype size (G ), and
carrying capacity (K ) (Table 1). Our simulations were
qualitatively invariable for the parameters G and K - only
results in varying b , μ , N0, andd0 are presented.