### Models

We follow Křivan and Diehl [

8] who considered the IGP module in which the dynamics of the IGpredator density

*P*, IGprey density

*N*, and resource density

*R*, are described by

$\frac{\text{d}R}{\text{d}t}=R\left(r\left(1-\frac{R}{K}\right)-\frac{{\lambda}_{\mathit{RN}}N}{1+{h}_{\mathit{RN}}{\lambda}_{\mathit{RN}}R}-\frac{{u}_{\mathit{RP}}{\lambda}_{\mathit{RP}}P}{1+{u}_{\mathit{RP}}{\lambda}_{\mathit{RP}}{h}_{\mathit{RP}}R+{u}_{\mathit{NP}}{\lambda}_{\mathit{NP}}{h}_{\mathit{NP}}N}\right)$

(1)

$\frac{\text{d}N}{\text{d}t}=N\left(\frac{{e}_{\mathit{RN}}{\lambda}_{\mathit{RN}}R}{1+{h}_{\mathit{RN}}{\lambda}_{\mathit{RN}}R}-\frac{{u}_{\mathit{NP}}{\lambda}_{\mathit{NP}}P}{1+{u}_{\mathit{RP}}{\lambda}_{\mathit{RP}}{h}_{\mathit{RP}}R+{u}_{\mathit{NP}}{\lambda}_{\mathit{NP}}{h}_{\mathit{NP}}N}-{m}_{N}\right)$

(2)

$\frac{\text{d}P}{\text{d}t}=P\left(\frac{{e}_{\mathit{RP}}{u}_{\mathit{RP}}{\lambda}_{\mathit{RP}}R+{e}_{\mathit{NP}}{u}_{\mathit{NP}}{\lambda}_{\mathit{NP}}N}{1+{u}_{\mathit{RP}}{\lambda}_{\mathit{RP}}{h}_{\mathit{RP}}R+{u}_{\mathit{NP}}{\lambda}_{\mathit{NP}}{h}_{\mathit{NP}}N}-{m}_{P}\right)$

(3)

where *r* is the intrinsic growth rate of the resource, *K* is the carrying capacity, *λ*_{
ij
} is the encounter rate of species *j* for species *i* *e*_{
ij
} is the efficiency of converting energy of species *i* for species *j* *h*_{
ij
} is the handling time of species *j* for species *i*, and *m*_{
i
} is the density-independent mortality rate of species *i*. *u*_{
iP
} is the probability that the IGpredator attacks species *i* upon an encounter. The fixed behavior model assumes that *u*_{
NP
} = 1 and *u*_{
RP
} = 1 (i.e., the IGpredator always attacks the IGprey and the resource).

Křivan and Diehl [8] considered that the IGpredator optimally chooses its prey according to a prey choice model [23]. The solution to this problem is well known [24] and is as follows. Suppose the IGprey is more profitable than the resource (*e*_{
NP
}/*h*_{
NP
} > *e*_{
RP
}/*h*_{
RP
}), the IGpredator always attacks the IGprey (i.e., *u*_{
NP
} = 1) when it encounters an IGprey. The IGpredator also always attacks the resource upon an encounter (i.e., *u*_{
RP
} = 1) when the the IGprey density is below a threshold density *N*_{
T
} = *e*_{
RP
}/(*λ*_{
NP
}*h*_{
NP
}*h*_{
RP
}(*e*_{
NP
}/*h*_{
NP
}*e*_{
RP
}/*h*_{
RP
})) but always ignores the resource (i.e., *u*_{
RP
} = 0) otherwise. Similarly, when the resource is more profitable than the IGprey (*e*_{
RP
}/*h*_{
RP
} > *e*_{
NP
}/*h*_{
NP
}), the IGpredator always attacks the resource (i.e., *u*_{
RP
} = 1) and also always attacks the IGprey (i.e., *u*_{
NP
} = 1) only when the resource density is below a threshold density *R*_{
T
} = *e*_{
NP
}/(*λ*_{
RP
}*h*_{
RP
}*h*_{
NP
}(*e*_{
RP
}/*h*_{
RP
}*e*_{
NP
}/*h*_{
NP
})) and entirely reject the IGprey (i.e., *u*_{
NP
} = 0) when the resource density is above the threshold density (*R* > *R*_{
T
}).

We consider a model which incorporates individual variation in the prey choice behavior where IGpredators have variable perceptions about the densities of the interacting species. In our model, individual IGpredators do not show partial preference [12, 25, 26]. Suppose only a fraction of IGpredators attack the resource at a given condition. In our model, this occurs because some IGpredators always attack the resource while the rest always ignore the resource (i.e., individual variation). On the other hand, in partial preference models, this occurs because all IGpredators attack the resource with the probability equals to the observed fraction of IGpredators that are attacking the resource (i.e., no individual variation).

The optimal behavioral expression (the all-or-nothing behavior determined by

*u*_{
RP
} and

*u*_{
NP
}) depends on the density of the more profitable prey of the two, and thus perceptual variation in the density leads to variable behavioral expressions among IGpredators [

27]. Because perceived densities take non-negative continuous values, we use a gamma distribution gamma(

*α* *β*) to describe their distribution. By specifying the mean

*μ* and variance

*σ*^{2} of the distribution,

*α* and

*β* can be described as

*α* =

*μ*^{2}/

*σ*^{2} and

*β* =

*σ*^{2}/

*μ*. We assume that the mean is the true density (e.g.

*μ* =

*N* if

*e*_{
NP
}/

*h*_{
NP
} >

*e*_{
RP
}/

*h*_{
RP
}). We also assume that the perceptional variance is the same as the mean (

*σ*^{2} =

*μ*). Then the dynamics of the IGP module with individual variation can be described by,

$\frac{\text{d}R}{\text{d}t}=R\left(r\left(1-\frac{R}{K}\right)-\frac{{\lambda}_{\mathit{RN}}N}{1+{h}_{\mathit{RN}}{\lambda}_{\mathit{RN}}R}-\frac{{\lambda}_{\mathit{RP}}P{q}_{\mathit{RN}}}{1+{\lambda}_{\mathit{RP}}{h}_{\mathit{RP}}R+{\lambda}_{\mathit{NP}}{h}_{\mathit{NP}}N}-\frac{{\lambda}_{\mathit{RP}}P{q}_{R}}{1+{\lambda}_{\mathit{RP}}{h}_{\mathit{RP}}R}\right)$

(4)

$\frac{\text{d}N}{\text{d}t}=N\left(\frac{{e}_{{}_{\mathit{RN}}}{\lambda}_{\mathit{RN}}R}{1+{h}_{\mathit{RN}}{\lambda}_{\mathit{RN}}R}-\frac{{\lambda}_{\mathit{NP}}P{q}_{\mathit{RN}}}{1+{\lambda}_{\mathit{RP}}{h}_{\mathit{RP}}R+{\lambda}_{\mathit{NP}}{h}_{\mathit{NP}}N}-\frac{{\lambda}_{\mathit{NP}}P{q}_{N}}{1+{\lambda}_{\mathit{NP}}{h}_{\mathit{NP}}N}-{m}_{N}\right)$

(5)

$\frac{\text{d}P}{\text{d}t}=P\left({q}_{R}\frac{{e}_{\mathit{RP}}{\lambda}_{\mathit{RP}}R}{1+{\lambda}_{\mathit{RP}}{h}_{\mathit{RP}}R}+{q}_{N}\frac{{e}_{\mathit{NP}}{\lambda}_{\mathit{NP}}N}{1+{\lambda}_{\mathit{NP}}{h}_{\mathit{NP}}N}+{q}_{\mathit{RN}}\frac{{e}_{\mathit{RP}}{\lambda}_{\mathit{RP}}R+{e}_{\mathit{NP}}{\lambda}_{\mathit{NP}}N}{1+{\lambda}_{\mathit{RP}}{h}_{\mathit{RP}}R+{\lambda}_{\mathit{NP}}{h}_{\mathit{NP}}N}-{m}_{P}\right)$

(6)

where

*q*_{
RN
}*q*_{R,}*q*_{
N
} are the fraction of IGpredators that attacks both the resource and the IGprey, the resource only, and the IGprey only, respectively. For example, when the IGprey is more profitable than the resource,

*q*_{
R
} = 0.

*q*_{
RN
} depends on the perceived density

*x* of the profitable prey

*N* and is,

${q}_{\mathit{RN}}={\int}_{0}^{{N}_{T}}f\left(x\right)dx$

(7)

where

*f*(

*x*) is the gamma distribution discussed above. In other words,

*q*_{
RN
} is the proportion of IGpredators that perceives the density of the IGprey is less than the threshold density

*N*_{
T
}. Because individual IGpredators either perceive that the density of the IGprey is greater than the threshold or not (i.e., one or the other), the proportion of IGpredators that perceive that the density of the IGprey is above the threshold (

*q*_{
N
}) is 1 −

*q*_{
RN
}. Similarly, when the resource is more profitable (which leads to

*q*_{
N
} = 0), and the proportion of IGpredators that perceives that the density of the resource is less than the threshold density,

*R*_{
T
} is,

${q}_{\mathit{RN}}={\int}_{0}^{{R}_{T}}f\left(x\right)dx$

(8)

followed by *q*_{
R
} = 1 − *q*_{
RN
}*.*

The effect of adaptive behavior and individual variation on coexistence is examined using invasion and stability analyses supplemented with numerical simulations. The parameter values used in the analyses follows a previous study for comparison [8]: *r* = 0.3, *λ*_{
RN
} = 0.037, *λ*_{
RP
} = 0.025, *λ*_{
NP
} = 0.025, *h*_{
RN
} = 3, *h*_{
RP
} = 4, *h*_{
NP
} = 4, *e*_{
RN
} = 0.6, *e*_{
RP
} = 0.36, *e*_{
NP
} = 0.6, *m*_{
N
} = 0.03, *m*_{
P
} = 0.0275.

### Invasion analysis

In this invasion analysis, we examine whether the IGpredator can invade communities that consist of the IGprey and resource, and whether the IGprey can invade communities that consist of the IGpredator and resource. When both the IGpredator and IGprey can invade each other (i.e., mutually invasible), coexistence is implied.

Suppose when resident communities are at equilibrium, the invasion conditions for the IGprey and IGpredator, respectively, are,

$\frac{\text{d}N}{\text{d}t}{\frac{1}{N}|}_{N=0}=\frac{{e}_{\mathit{RN}}{\lambda}_{\mathit{RN}}{R}_{\mathit{RP}}^{*}}{1+{h}_{\mathit{RN}}{\lambda}_{\mathit{RN}}{R}_{\mathit{RP}}^{*}}-\frac{{u}_{\mathit{NP}}{\lambda}_{\mathit{NP}}{P}_{\mathit{RP}}^{*}}{1+{u}_{\mathit{RP}}{\lambda}_{\mathit{RP}}{h}_{\mathit{RP}}{R}_{\mathit{RP}}^{*}}-{m}_{N}$

(9)

$\frac{\text{d}P}{\text{d}t}{\frac{1}{P}|}_{P=0}=\frac{{u}_{\mathit{RP}}{\lambda}_{\mathit{RP}}{e}_{\mathit{RP}}{R}_{\mathit{RN}}^{*}+{u}_{\mathit{NP}}{\lambda}_{\mathit{NP}}{e}_{\mathit{NP}}{N}_{\mathit{RN}}^{*}}{1+{u}_{\mathit{RP}}{\lambda}_{\mathit{RP}}{h}_{\mathit{RP}}{R}_{\mathit{RN}}^{*}+{u}_{\mathit{NP}}{\lambda}_{\mathit{NP}}{e}_{\mathit{NP}}{N}_{\mathit{RN}}^{*}}-{m}_{P}$

(10)

where

${R}_{\mathit{RP}}^{*}$ and

${P}_{\mathit{RP}}^{*}$ are the equilibrium densities of the resource and IGpredator in the resouce-IGpredator community;

${R}_{\mathit{RN}}^{*}$ and

${N}_{\mathit{RN}}^{*}$ are the equilibrium densities of the resource and the IGprey in the resource-IGprey community, respectively. When the expressions in Equations (

9) and (

10) are both positive, mutually invasibility is established. When resident communities exhibit cycles, the invasibility conditions of the IGprey and IGpredator, respectively, are,

$\frac{1}{{\tau}_{\mathit{RP}}}{{\int}_{0}^{{\tau}_{\mathit{RP}}}\frac{\text{d}N}{dt}\frac{1}{N}|}_{N=0}dt$

(11)

$\frac{1}{{\tau}_{\mathit{RN}}}{{\int}_{0}^{{\tau}_{\mathit{RN}}}\frac{\text{d}P}{dt}\frac{1}{P}|}_{P=0}dt$

(12)

where τ_{
RP
} and τ_{
RN
} are the periodicity of the cycle for the resource-IGpredator and the resource-IGprey resident communities, assuming resident populations are on the trajectory of the limit cycles.

The inclusion of individual variation does not affect results of the invasion analysis. The possibility of invasion of the IGprey is not affected by individual variation because the perceptual variance is the same as the true density (e.g., at an invasion event, the variance is 0). The possibility of invasion of the IGpredator is also unaffected by individual variation if we assume that some invading individuals exhibit optimal behavior.