Y. Ren et al. / Applied Mathematics and Computation 323 (2018) 95–105 97
Table 1
Summary of notions.
Symbol Meaning
N The number of sites of a square lattice
s
i
The strategy of individual i
b The payoff of a defector when playing a PDG with a cooperator
ρ The density of population in a square lattice
P
l
i
The expected payoff of individual i in site l
p
i → j
The payoff of individual i when playing a PDG with individual j
n
l
C
The number of cooperative neighbors in site l
n
l
D
The number of defective neighbors in site l
M The Moore neighbor range size of a square lattice
F P
l
i
The fairness payoff of individual i in site l
P
l
i
The average expected payoff of individual i in site l
P
l
i
The average payoff of individual i ’s neighbors
α The fairness factor
i
The set of individual i ’s neighbors
|
i
| The number of individual i ’s neighbors
f
c
The fraction of cooperators
r The mutation rate
payoff parameter. The payoff matrix is
C D
C
D
1 0
b 0
(1)
Initially, N individuals are randomly located on a square lattice of L × L sites with periodic boundary conditions. Each
site in the lattice can be either empty or occupied by one individual. The population density is ρ = N/L
2
. Individuals are
updated asynchronously in a random sequential order so that N iterations can be regarded as one generation ( g ) [37] . Each
iteration consists of three steps, namely: interaction , migration and imitation . These three steps are also known as combat ,
diffusion and offspring (i.e., CDO dynamics) in previous studies such as [17,19] .
2.1. Interaction
During the interaction step, each individual interacts with its Neumann neighbors (i.e., individuals in four direct-linked
sites) and obtains an accumulated payoff as its benefit in this iteration. For individual i , the expected payoff P
l
i
in site l is
calculated as
P
l
i
=
j∈
i
p
i → j
, (2)
where p
i → j
is the payoff of i when interacting with j .
i
represents the neighbor set of i . According to the payoff matrix in
Eq. (1) , Eq. (2) can be re-written as
P
l
i
=
n
l
C
, s
i
= 1 ;
n
l
C
· b, s
i
= 0 .
(3)
Here, n
l
C
is the number of cooperative neighbors of i in site l .
2.2. Migration
In the migration step, success-driven migration individuals explore the expected payoff for all empty sites in Moore
neighborhood of range M [17] , i.e., the square region with size of (2 M + 1)
2
[9] . If both cooperators and defectors migrate
to the site with a higher payoff, then it may cause cooperative clusters to be surrounded by defectors [20] . To solve this
problem, we consider the fairness factor and introduce a neighbor-considered migration in our model, where individuals
pursue higher payoff as well as reducing the disparity of payoff among their neighbors. Therefore, instead of expected
payoff, we define a fairness payoff in our model. For each individual i , its fairness payoff F P
l
i
can be denoted as follows
F P
l
i
= α · P
l
i
+ (1 − α) · (−| P
l
i
− P
l
i
| ) , (4)
where P
l
i
is the average expected payoff of i , and P
l
i
is the average of payoff of i ’s neighbors. | P
l
i
− P
l
i
| is used
to measure the difference of average expected payoff between i and its neighbors. α(0 ≤ α ≤ 1) is a fairness weight factor
to balance the weight of its own benefit and neighbors’ benefit. For α = 1 and M = 1 , the fairness payoff is the same as
the expected payoff, and our strategy can be regarded as the success-driven migration strategy. For α = 0 , the difference