  The emergence and maintenance of cooperative behavior is a fascinating topic
in evolutionary biology and social science. The public goods game (PGG) is a
paradigm for exploring cooperative behavior. In PGG, the total resulting payoff
is divided equally among all participants. This feature still leads to the
dominance of defection without substantially magnifying the public good by a
multiplying factor. Much effort has been made to explain the evolution of
cooperative strategies, including a recent model in which only a portion of the
total benefit is shared by all the players through introducing a new strategy
named persistent cooperation. A persistent cooperator is a contributor who is
willing to pay a second cost to retrieve the remaining portion of the payoff
contributed by themselves. In a previous study, this model was analyzed in the
framework of well-mixed populations. This paper focuses on discussing the
persistent cooperation in lattice-structured populations. The evolutionary
dynamics of the structured populations consisting of three types of competing
players (pure cooperators, defectors and persistent cooperators) are revealed
by theoretical analysis and numerical simulations. In particular, the
approximate expressions of fixation probabilities for strategies are derived on
one-dimensional lattices. The phase diagrams of stationary states, the
evolution of frequencies and spatial patterns for strategies are illustrated on
both one-dimensional and square lattices by simulations. Our results are
consistent with the general observation that, at least in most situations, a
structured population facilitates the evolution of cooperation. Specifically,
here we find that the existence of persistent cooperators greatly suppresses
the spreading of defectors under more relaxed conditions in structured
populations compared to that obtained in well-mixed population.
