Figure 1.
The schematic diagram for the ecological models.
(A)Predation model. (B)Competition model. (C)Mutualism model. The potential landscape is linked with the probability
by
in species space. (D) Limit cycle attractor. The barrier height from the maximum inside the closed ring to the potential maximum along the ring can quantify the stability of the limit cycle attractor. (E) Multiple attractors. There are four stable states: survival alone state
of species
, survival alone state
of species
, coexisting state
, and both extinct state
.
is the saddle points between the attractors
and
while
is the saddle points between the attractors
and
. The barrier heights from the saddle points to the potential minimums of the basins can quantify the stability of each attractor.
Figure 2.
The potential landscapes for the predation, competition and mutualism models.
Top row: The population potential landscape ((A) predation model. (B) competition model. (C) mutualism model.) Purple arrows represent the flux velocity(
) while the black arrows represent the negative gradient of population potential(
). Bottom row: The potential intrinsic energy landscape
. ((D) predation model. (E) competition model. (F) mutualism model.). Purple arrows represent the intrinsic flux velocity(
) while the black arrows represent the negative gradient of intrinsic potential(
).
Figure 3.
The barrier height of population landscape, escape time and dissipation rate versus the diffusion coefficient for predation model.
(A) The barrier height of the population landscape versus the diffusion coefficient
. (B) The escape time versus the barrier height of population landscape. (C) The dissipation rate versus the diffusion coefficient
.
Figure 4.
The barrier height of the population landscape, escape time and dissipation rate versus the rate parameters for predation model.
(A) The barrier height of the population landscape versus . (B) The escape time versus barrier height of the population landscape for changing
. (C) The dissipation rate versus
. (D) The barrier height of the population landscape versus
. (E) The escape time versus barrier height of the population landscape for changing
. (F) The dissipation rate versus
. (G) The barrier height of the population landscape versus
. (H) The escape time versus barrier height of the population landscape for changing
. (I) The dissipation rate versus
.
Figure 5.
The barrier height of intrinsic potential landscape and free energy versus the rate parameters for predation model.
The barrier heights of intrinsic potential landscape versus parameters (A),
(B),
(C). The free energy versus
(D),
(E),
(F).
Figure 6.
The coherence for predation model.
(A)The coherence versus the diffusion coefficient. (B) The coherence versus the parameter . (C) The coherence versus the the parameter
. (D) The coherence versus the the parameter
.
Figure 7.
The barrier height of the population landscape, escape time and dissipation rate versus the diffusion coefficient for competition model.
(A) The barrier height of the population landscape versus the diffusion coefficient
. (B) The escape time versus the barrier height of the population landscape. (C) The dissipation rate versus the diffusion coefficient
.
Figure 8.
The barrier height of the population landscape, escape time and dissipation rate versus the rate parameters for competition model.
(A) The barrier height of the population landscape versus . (B) The escape time versus barrier height of the population potential landscape. (C) The dissipation rate versus
. (D) The barrier height of the population potential landscape versus
. (E) The escape time versus barrier height of the population potential landscape. (F) The dissipation rate versus
. (G) The barrier heights of the population landscape versus
. (H) The escape time versus barrier height of the population potential landscape. (I) The dissipation rate versus
.
Figure 9.
The barrier height of intrinsic potential landscape and free energy versus the rate parameters for competition model.
The barrier heights of intrinsic potential landscape versus (A),
(B),
(C). The free energy versus
(D),
(E),
(F).
Figure 10.
The barrier height of the population landscape, escape time and dissipation rate versus the diffusion coefficient for mutualism model.
(A) The barrier height of the population landscape versus the diffusion coefficient
. (B) The escape time versus the barrier height of the population landscape. (C) The dissipation rate versus the diffusion coefficient
.
Figure 11.
The barrier height of the population landscape, escape time and dissipation rate versus the rate parameters for mutualism model.
(A) The barrier heights of the population landscape versus . (B) The escape time versus barrier height of the population landscape for changing
. (C) The dissipation rate versus
. (D) The barrier height of the population landscape versus
. (E) The escape time versus barrier height of the population landscape for changing
. (F) The dissipation rate versus
. (G) The barrier height of the population landscape versus
. (H) The escape time versus barrier height of the population landscape for changing
. (I) The dissipation rate versus
.
Figure 12.
The barrier height of intrinsic potential landscape and free energy versus the rate parameters for mutualism model.
The barrier heights of intrinsic potential landscape versus (A),
(B),
(C). The free energy versus
(D),
(E),
(F).
Figure 13.
The paths for competition model and mutualism model.
(A) The paths on the intrinsic potential landscape for competition model. (B) The paths on the intrinsic potential landscape
for mutualism model. (C) The paths on the population potential landscape
for competition model. (D) The paths on the population potential landscape
for mutualism model.
Figure 14.
The potential landscape, barrier height of the population landscape and the sensitivity of parameters for lynx-snowshoe hare model.
(A) The population potential landscape for lynx-snowshoe hare model. (B) The barrier heights versus changing parameters. The basic set of the parameters are: . (C) The barrier heights versus the hares' rate of population growth.