|
nif:isString
|
-
2.1 Geography of Ontario Parks: The interactions between forest pest invasions and human decisions regarding firewood transportation can be better understood in the context of the spatial distribution of Ontario Parks (the provincial park system of Ontario, Canada). The spatial distribution of parks and the strength of connections between parks can have a large impact on cross–park (“cross–patch”) infestation spread. Ontario Parks can be divided into two categories: operating parks that are regulated under the provincial authority, and non–operating parks that have no fees/staff and only limited facilities. According to Ontario Parks statistics from 2010, the southern and central regions of Ontario are the most popular places for day–use visitors and overnight campers [19]. These regions have 75% to 80% of the total visitors of all Ontario parks per year [19]. Large populations visit the central region of Ontario, and these may include visitors from northern and southern regions. In research on the attitudes of visitors regarding use of left-over (residual) firewood at Wisconsin State parks, it was found that visitors take up to 15% of unused firewood back to their homes [20]. If this firewood is infested it could become the seed of a new local infestation. Ontario Parks are numerous and highly connected to one another, presenting a complicated geometrical structure [21]. With respect to the movement of camp firewood, the parks are connected through visits from the public. The high connectivity between parks stems from the fact that any given park may receive visitors who reside anywhere in Ontario, any of whom may bring along firewood that was purchased at their point of origin. Such high connectivity can be approximated by a mesh topology (Fig 1). The parks in the central region of Ontario are most dense and experience the highest volume of visitors. The southeastern region has a high density of parks as well but the visitor volume is less than the visitor volume of central parks. The parks in these regions are located mostly inland. In contrast, the southwestern parks distributed linearly along lakesides have a slightly higher volume of visitors than southeastern parks. Northern parks exhibit the same tendency for inland locations however the density of parks and the volume of visitors are significantly lower than the parks in central and southern regions.
Figure data removed from full text. Figure identifier and caption: 10.1371/journal.pone.0139353.g001 Patch geometry used for the simulation model.Partition 1 patches experiences a high visitor volume while Partition 2 patches experiences a low visitor volume. Transport strategists move firewood between patches in both partitions.
Here, we simplify this structure to capture the basic distinction between centrally located parks with high visitor volumes, and proximal parks with low visitor volumes. We divide Ontario parks into two partitions: one for the southern and central Ontario, which receives a high volume of visitors, and the other for Northern Ontario, which receives a low volume of visitors. Each patch in either partition is connected with all patches in that partition as well as with all patches in the other partition as shown in Fig 1. Hence, the total system is comprised of completely connected patches while patches are placed in either of the partitions according to their volume of visitors. Moreover, the strength of connection varies depending on whether patches are in the high volume or low volume partition: patches in the low volume partition receives a lower volume of visits from other patches in the same partition as well as a lower volume of visits from patches in the high volume partition.
We developed a discrete–time, mechanistic, stochastic model to study pest invasions, human decision–making, and firewood movement in the 10–patch system. The simulation time step is one week. Each patch has a population of Si(t) susceptible trees and Ii(t) infested trees. The total carrying capacity of susceptible and infested trees is K. Susceptible trees become infested trees according to specific transition probabilities based on assumed transmission mechanisms (details are below). Susceptible trees become infested with a given probability per time step due to local infestation (infestation within a patch) according to “standard incidence” transmission mechanism. Susceptible trees also become infested due to firewood transport from other infested patches, depending on the prevalence of infestation in the other patch, visitor volumes between the patches, and human decisions regarding whether or not to transport firewood between the patches. Infested trees die with a certain probability per time step, and in the model notation use ‘R’ to denote a ‘removed’ state, for trees that have died after being infested. Each patch is considered to include both a local park as well as the local residents of that area. Hence, the individuals who live in patch i can either be local strategists (where Li(t) represents the proportion of local strategists in patch i) who always buy and burn firewood at the patches they visit, or they can be transport strategists (where Ti(t) represents the proportion of transport strategists in patch i) who always bring firewood from their home patch i when they visit one of the other 9 patches (and thus risk spreading the infestation, if patch i is infested). Thus, we can write the relation between the proportion of local and transport strategists as Li(t) = 1 –Ti(t). In particular, the probability per time step that a susceptible tree becomes infested due to local (within–patch) dispersal is PS to I,i(t)=βIi(t)K(1) where β is the transmission probability constant, Ii(t) is the number of infested trees in patch i, and K is the carrying capacity in patch i (Table 1). This form of the transmission probability, where the infestation rate depends on the density of infested trees, is referred to as standard incidence [22]. We note that the time dependence of PS to I,i occurs due to its dependence on the variable Ii(t), representing the number of infested trees at time t. The nonlinearity of the simulation model stems from Eq 1, since the probability of infestation PStoI,i depends on the infestation prevalence Ii(t)/K instead of being a constant value, therefore the mean of the binomial distribution for the number of trees that get infected depends on the nonlinear term Si(t)Ii(t)/K.
Table data removed from full text. Table identifier and caption: 10.1371/journal.pone.0139353.t001 Model variables and transition probabilities. The total number of susceptible trees that become infested per time step, numStoI(t), can be found from the probability PStoI(t) by sampling from a binomial distribution according to numS to I(t)=Binomial(Si(t),PS to I(t)),(2) Once a susceptible tree becomes infested, it survives for an average time duration D. Thus, an infested tree has a probability PI to R=ε=1D(3) per time step of dying. Thus, the number of trees numI to R that die in each time step can be written as numI to R(t)=Binomial(Ii(t),PI to R),(4) New trees can grow in the place of dead trees. However, their growth is subject to the availability of space and the growth rate (fecundity) r of susceptible trees. Therefore, if we assume that (Ii(t) + Si(t)) / K is the proportion of trees already existing in the patch, then 1 –(Ii(t) + Si(t)) / K would be the available space for new trees to grow. Therefore, the probability P0 to S that a susceptible tree gives rise to a (susceptible) offspring is P0to S(t)=r(1−Ii(t)+Si(t)K)(5) Hence, the number of new trees per time step is num0to S(t)=Binomial(Si(t),P0to S(t)),(6) Firewood cost, the influence of pest outbreaks upon decision–making, social norms and imitation dynamics (e.g. social learning) inform decision–making. If the local firewood cost in a patch is higher than the cost of bringing firewood then the individuals visiting from other patches may bring their own firewood instead. However, individuals are also influenced by awareness of the impact of pest outbreaks in their own patch, and emerging rules about social conduct [10]. These social, financial and behavioral factors are together used to define the utility (a quantitative measure of preference) for these strategists, representing the utility for being a local strategist who always “burns where they buy” versus the utility for being a transport strategist who transports firewood before burning it. In particular, we define the utility for a local strategist as UL(t) while the utility for a transport strategist as UT(t). The equations for UL(t) and UT(t) are UL(t)=−CL+n(Li(t)−0.5)UT(t)=−CT+n(0.5−Li(t))−fIi(t)}(7) where CL and CT are the local and transport firewood costs respectively and n controls the strength of social norms. The parameter n can be interpreted as the strength of social pressure in favor of the dominant behavior or attitude [23,24]. If n is large and there are many local strategists (Li(t) is high), then UL(t) is high since there is social pressure for individuals to conform to the norm of not transporting firewood, and this tends to further increase Li(t). Conversely, if Li(t) is low, causing UT(t) to be high, then social pressure will further reduce Li(t). The infestation concern parameter f is a proportionality constant that controls the extent to which infestation prevalence influences individual decision-making. The total number of individuals in a patch is a constant NS = NL(t) + NT(t), where NL(t) is the number of local strategists and NT(t) is the number of transport strategists. Hence, the total number of individuals remains the same throughout the simulation. Accordingly, Li(t)=NL,i(t)NS,iandTi(t)=NT,i(t)NS,i(8) We assume that an individual “samples” (i.e. speaks to) other persons in their own patch regarding firewood transport, firewood cost, and forest pest infestations with a specific probability per unit time [10,25]. Sampling may occur through person-to-person contact or through other means such as social media or telephone. During this interaction, individuals compare their utilities received for their respective strategies. If a sampled person is playing a different strategy and is receiving a higher utility, then the individual doing the sampling switches to the sampled person’s strategy with a probability proportional to the expected gain in utility. Therefore, the total probability per time step that a local strategist becomes a transport strategist is the product of the probability of sampling (the “social learning rate”) and the probability of switching strategies: PL to T(t)={0UL(t)≥UT(t)σ(UT(t)−UL(t))UL(t)UT(t)(10) The number of individual changing strategies in each time step is then numL to T(t)=Binomial(NL,i(t),PL to T(t)),(11) numT to L(t)=Binomial(NT,i(t),PT to L(t)),(12) Just as infestation influences human behavior, human behavior regarding firewood transportation influences infestation spread between patches. The probability that a susceptible tree in a given patch gets infested due to transported firewood depends upon the proportion of transport strategists in neighboring infested patches and the amount of travel (and thus firewood movement) between patches. Thus the probability of cross–patch infestation occurring in patch i due to infested firewood from patch j can be written as, PS to IT,j(t)=βTi(t)Ii(t)K{dHi=1..5(within P1)dLOtherwise(within P2and between P1and P2),i≠j(13) where dH > dL since the volume of visitors within Partition 1 (P1) is significantly higher than either the volume of visitors within Partition 2 (P2) or the volume of visits between the two partitions. Based on these transmission probabilities, the number of new infestations in patch i per time step due to cross–patch movement of firewood is the sum of all cross–patch infestations introduced from patches j ≠ i: numS to IT,i(t)=∑j≠iBinomial(Si(t),PS to IT,j(t)),(14) Cross–patch infestation can occur several times during the simulation. However, the first cross–patch infestation event is most important since it forms the nucleus of the first outbreak in a previously un-infested patch. Thus, an important outcome variable is time–to–first–cross–patch–infestation, defined as the time between the start of the simulation and the time that a patch first becomes infested due to firewood transport from another patch. Once the number of state transitions is computed from the binomial sampler at each time step from Eqs 2, 4, 6, 11, 12 and 14, the state variables Si(t), Ii(t), Li(t), and Ti(t) are updated. The number of susceptible trees increases with the growth of new susceptible trees and decreases due to infestations originating either inside the patch or from other patches, thus: Si(t+1)=Si(t)−numS to I,i(t)+num0to S,i(t)−numS to IT,i(t)(15) The infested trees increase correspondingly increase due to spread of infestation, but their number decreases when infested trees die: Ii(t+1)=Ii(t)+numS to I(t)−numI to R(t)+numS to IT(t)(16) Finally, transitions between the numbers of local and transport strategists are given by NL,i(t+1)=NL,i(t)−numL to T(t)+numT to L(t)(17) NT,i(t+1)=NT,i(t)+numL to T(t)−numT to L(t)(18) Baseline parameter values were obtained from empirical data concerning tree species, forest pest infestations, and firewood costs (Table 2). Many parameter values were borrowed from the previous 2-patch deterministic model [10]. The ecological parameters, i.e. fecundity of trees r, differs for various trees species infested by EAB and ALB, while the probability of transmission β and mortality probability per time step of infested trees ε depends upon the density of insects in the patch. These parameters are well studied in the literature [7,10,11,19,26–28]. We used the empirical results found in [26] to estimate the fecundity of trees r = 0.06 / year by multiplying the mean survival of trees species, i.e. 2.4 ·10−6 trees / seed, with the total number of seeds · tree-1 · year-1, i.e. 25,000. The transmission probability parameter β was determined by a simple spread model to approximate the insect’s arrival time in a patch. Various spread rates have been estimated in the literature, ranging from low, i.e. 10 km/year, to high, i.e. 50 km/year, [28,29]. The parameter was determined by adjusting the distance covered by insects according the area covered. We assumed 5 ha = 50000 m2 of land with a spread rate of 25000 m2 / year, yielding the transmission probability parameter β = 0.5 / year. The fatality probability ε = 1/3 per year was taken from literature [27] stating that it takes three years, on average, for a tree infested by EAB to die in recent infestations in Michigan and Ontario. The economic parameters for the cost of local and transported firewood, CL and CT respectively, vary by region. However, in the case of Ontario, baseline values for the parameters CL = $ 6.75 and CT = $ 5.00 were determined by surveying park administration offices and surveying the local markets [10].
Table data removed from full text. Table identifier and caption: 10.1371/journal.pone.0139353.t002 Parameter definitions and their baseline values. System–specific empirical data were not available for some sociological parameters (e.g. n, f, σ) hence baseline values for these parameters were calibrated until ecologically and sociologically plausible dynamics were obtained (Table 2). In particular, we sought model dynamics consistent with recent outbreaks with EAB and ALB in Ontario and other jurisdictions: infestation generates some level of concern in the population, but the concern is not sufficient to prevent regional spread, which occurs on the timescale of several years. Moreover, infestation spreads more quickly to patches that experience a high volume of visitors than patches that experience a low volume of visitors. Univariate sensitivity analysis was used to study the impact of varying social influence n, social learning σ, outbreaks f, fatality rate ε, firewood cost CL and controlling rate d, by changing these parameters one at a time while the other parameters remained at their baseline values.
|
![[This page as RDF]](http://data.gesis.org/somesci/static/rdf-icon.gif){kind=icon}