A model of human hunting pressure on a population of male and female animals in a possibly variable and dynamic environment.
The MHPM divides the prey species into male and female animals that are modeled independently to allow for sexual dimorphism and the possibility that hunters will prefer one over the other. Both foragers and prey animals have some (limited and biased) information about local conditions that they use to plan their movement and subsistence decisions. Forager decisions are still based on optimality logic, but the information in the decision models is more realistic than in classic numerical OFT modeling. There are feedback loops connecting the consequences of all these decisions in a linked complex adaptive social-ecological system that create temporal and spatial dynamics not typically incorporated into OFT modeling.
The MHPM divides the prey species into male and female animals that are modeled independently to allow for sexual dimorphism and the possibility that hunters will prefer one over the other. In classic OFT, the payoff (or "profitability"), Pi , of a potential prey species i is described as the food energy, Ei, gained (e.g., in kcal) per unit time, ti, spent "processing" the prey once encountered (this includes the time to pursue, process, and eat the animal):
Pi = Ei / ti
(1)
Alternatively, profitability may be calculated as food energy, Ei, gained minus energy costs expended handling, hi, of the prey item so that the profitability equation takes the form:
Pi = Ei - hi
(2)
To establish a "Diet Breadth" model, the profitability of all potential prey species must be known, as well as the average "search cost," S, required to find an individual prey animal of each species. To be included in the diet, the profitability of prey species k must exceed or equal the potential profitability of all other higher-ranked prey species (i to j) when search costs are factored in:
if: Pk PiPj - Sk, then include animal k in diet
(3)
Thus, if a first-ranked prey animal is encountered it is always pursued, processed, and eaten. If a lower-ranked prey animal is encountered, it is only pursued, processed, and eaten if the densities of all higher-ranked animals are low enough to offset the reduced payoff offered by the lower-ranked animal. This logic assumes the forager to know a priori the search time for all prey species within the entirety of the foraging environment, to always estimate these values correctly, and to never deviate from this decision logic. These issues are often cited as weaknesses of classical OFT, and the rebuttal is that OFT models are simply internally consistent baseline hypotheses against which to evaluate foraging datasets. In the MHPM, we combine OFT with a CAS perspective to move beyond these debates. Specifically, we leverage the spatiality and time-transgressive nature of the ABM modeling formalism to incorporate the effects of error and bias into the OFT decision logic in a way that may more realistically simulate foraging decisions, while still allowing for optimality logic to be used as a baseline.
In our model, the foraging decision logic reflects a two-prey diet breadth model that also incorporates some assessment of potential future search costs in addition to prior "sunk" search costs. Forager agents move and search iteratively in a dynamic landscape populated by a population of mobile prey animal agents. The only two prey choices in the model are assumed to be males and females of the same species, which can be individually parameterized in the model interface with distinct intrinsic food energy payoff and handling costs. Optionally, an additional food energy value for calves can be added to that of pregnant or recently postpartum females still rearing their offspring, effectively producing a third prey choice:
Pmale = Emale - hmale
(4)
Pfemale = Efemale - hfemale
(5)
Pfemale with calf = Efemale + Ecalf - hfemale
(6)
At the instantiation of any model run, the above payoffs are calculated from the values set in the model interface, a prey ranking order is determined, and knowledge of these prey ranks and payoffs is passed to all forager agents. Forager and prey animals begin to move as time progresses in a series of discrete time "ticks". A forager agent is considered to have encountered a potential prey agent in a given tick when both agents occupy the same or immediately adjacent landscape cells. At the time of an encounter, the foraging agent conducts a diet breadth calculation to decide whether to process the encountered prey, or to continue to search for another prey item.
At each time step, the forager agents "scan" within a pre-set foraging radius, r, to attain the current count of animals, n, within the foraging radius. This is a proxy for the number of prey that are likely to be nearby, and thus reasonable to consider as plausible alternative prey choices when a specific prey animal is encountered. With this information, forager agents then estimate an average search cost, Sa, to keep looking for a nearby first-ranked prey by dividing the area within the search radius by the number of prey that are within it, and then multiplying that by the forager movement cost, m. In the model, the foraging radius is delineated as a discrete number of cells, A, that fall completely within a linear radius of r around (and including) the central cell occupied by the forager so that the scanning formula takes the form:
Sa = ( A / n ) m
(7)
Where:
``A ⪅ πr2`
(8)
The estimated average pay-off, Pa, of the animals (i to j) within the foraging radius is then calculated as:
Pa = PiPj /A
(9)
If the payoff of an encountered animal, Pk, is higher than the estimated return of searching for, encountering, and then processing one of the first ranked prey animals within the search radius, then the currently encountered lower-ranked animal is processed and eaten:
if: Pk ≥ Pa - Sa, then consume animal k
(10)
At each time step, the current energy states of the foragers, Ecurrent, are updated as a function of their previous energy state, Eprevious, the forager movement costs, mforager, and any new foraging payoffs, P:
Ecurrent = Eprevious - mforager + P
(11)
Foragers are considered to be experiencing extreme food stress if their current energy state, Ecurrent, is less than the estimated average search costs, Sa. When in extreme food stress, foragers will process and eat any prey animal they encounter. This accounts for "sunk" search costs and non-optimal foraging decisions when appropriate:
if: Ecurrent \< Sa, then consume any encountered animal
(12)
Finally, foragers can optionally be considered to have the ability to store surplus food. If this option is enabled, then any energy value gained from the processing of an encountered animal that is in excess of the maximum consumption capacity (i.e., in excess of 100 percent of forager energy) can still be consumed in the subsequent time steps. Foragers will consume this excess at the same rate as the movement cost per tick, but they will not move or attempt to forage until this excess is fully consumed.
Because the main purpose of the MHPM is to understand extinction dynamics, it includes a detailed prey animal demographic component. When a male and female prey agent occupy the same or adjacent landscape cells, they can mate and produce offspring. The probability of a successful mating is linked to the energy status of the female animal, where the chance of producing an offspring decreases linearly from 100% at a full energy state to 0% at a zero energy state. Further, only females that have not recently calved are able to reproduce. This is controlled by a birth spacing interval variable, Tbirth spacing, that is set in the user interface. When a female reproduces, a counter, Tr, is set to zero and increases with each successive time tick. If Tr is less than or equal to *Tbirth spacing *then the female is considered to be pregnant, and eventually post-partum, with a calf:
if: Tr ≤ Tbirth spacing, then female is with calf
(13)
OnceTr is greater than Tbirth spacing, the calf is considered to be full grown and will be weaned and live separately from its mother.
if: Tr \> Tbirth spacing, then calf is weaned
(14)
Within the model, weaning spawns a new prey agent in the same landscape cell as the mother. The sex of the new prey agent is randomly determined with a 50% probability to be male or female, and it will be instantiated with the full adult food energy value and handling costs of the determined sex.
Births are balanced by a variety of causes of death. Any form of death for a postpartum female within the birth spacing window also includes the death of the calf. Firstly, any prey animals that are harvested by a forager in a time step are considered "killed" and are removed from the simulation environment. Prey animals may also die from a variety of "natural" causes such as starvation, old age, disease, or non-human predation. They are instantiated with initial energy states, E0, a typical age of senescence, Tsenescence, a maximum possible lifespan, Tmax, and must graze grass to survive. Each movement step (see main text, Section 4.2.3) reduces an animal's energy state by the animal movement cost, manimal, and each patch of grass encountered is grazed to replenish an animal's energy by the energy gain value, Egraze.
Ecurrent = Eprevious - manimal + Egraze
(15)
If there is no grass on the current patch, then the energy state of the animal is determined as:
Ecurrent = Eprevious - manimal
(16)
If an animal's energy state drops to 0 it will die:
if: Ecurrent ≤ 0, then animal dies
(17)
In every time step, there is a random chance of death among the population of animals with energy below a starvation threshold, Estarve. The chance of death is controlled by an internal mortality variable, pdeath:
for nanimals where: Ecurrent \< Estarve, a random selection of nanimals \* *pdeath* die
(18)
Likewise, if an animal reaches maximum lifespan, Tmax, they will die:
if: Tcurrent ≥ Tmax, then animal dies
(19)
But, as animals surpass the age of senescence, there is an increasing probability of death from old age within that population of senescent animals, again controlled by the internal rate of mortality, pinternal death:
for nanimals where: Tcurrent ≥ Tsenescence, and: Tcurrent \< Tmax, a random selection of nanimals \* pinternal death die
(20)
There is also a stochastic external mortality probability, pexternal death, that can be adjusted to simulate a percentage of animals that die from disease, accidents, or non-human predation at each time-step:
for nanimals, a random selection of nanimals \* pexternal death die
(21)
In each time step, animal ages are first increased by one. They then move to a new patch, and movement costs are deducted. They then graze grass if possible, and any energy gains are added. Then, deaths are assessed in the order of 1) human predation, 2) starvation risk, 3) old age risk, and finally, 4) external mortality risk. Next, mating occurs if conditions are correct. Finally, any female animals at the end of their birth spacing interval will wean their calves, which are spawned as new prey animal agents. Thus, at the end of each time step, the total new size of the population of prey animals is set for the next time step, and each animal has a newly updated set of energy and lifespan attributes.
Forager demography in the model is much simpler than animal demography: foragers may be either "immortal" or allowed to die if their energy state reaches zero:
if: Ecurrent ≤ 0, then forager dies
(22)
This simplicity is purposeful, as it allows controlled experimentation about the effect of human foraging pressure on prey population dynamics. If foragers are made "immortal," then hunting pressure is constant across time and the direct impact of human predation on extinction processes is easier to understand. If foragers are allowed to die of starvation, then hunting pressure may equilibrate with prey demographic processes over time, and more nuanced (but more difficult to follow) patterns in predator-prey dynamics may unveil.
We incorporate realistic movement algorithms for forager and animal agents. Forager agents are programmed to move with some foraging information feedback as local conditions change. At each time step, forager agents will turn to face the closest highest-ranked prey animal within their foraging radius, and then will advance one cell in that direction. If there are no nearby animals, then the forager proceeds in a true "random-walk" fashion by choosing a random heading and advancing to the nearest cell in that direction. For each movement, a predetermined movement cost is incurred.
Animal agents are programmed to instinctively move as a herd with some feedback about local grazing conditions. At each time step, animals will first attempt to face the nearest patch of grass within a 3-cell radius and advance one cell in that direction. If there is no grass in this radius, they will instead align their heading towards the average heading of animals that are within a 5-cell radius and advance one cell in that direction. If there are also no nearby animals, then the animal will proceed in a true "random-walk" fashion by advancing one cell in a randomly chosen direction. As above, for each movement, a predetermined movement cost is incurred.
An emergent property of these movement algorithms is that buffalo often self-organize into herds that will "migrate" in waves following the seasonal regrowth of grass patches (see main text, Section 4.2.4), with hunters following along, often in "bands" of multiple foragers, behind them.
Finally, the model incorporates several environmental dynamics components that can be switched on or off in any given experiment. These are controlled by a set of variables related to the spatial and temporal patterning of grass and grass regrowth. The basic spatial patterning of grass in the environment is determined by the grass proportion, pgrass, where the total number of grassed cells is determined by a random selection of ngrass cells from the total number of landscape cells, ntotal:
ngrass = pgrass \* ntotal
(23)
If grass regrowth is turned off, then the spatial patterning of grass never changes throughout a simulation run, and grass will always be available to be grazed in a grassed patch. Turning on grass regrowth enables initially grass-covered patches to become depleted when grazed by prey animals. These patches will remain depleted of grass for a user-determined number of time-steps, Tgrass regrowth, before the grass is replenished. In this situation, the current number of grassed patches, ngrass current, is determined by the following equation:
ngrass current = ngrass - ngrazed + nregrown
(24)
Where:
nregrown = ngrazed in time step {Tcurrent - Tgrass regrowth}
(25)
If regrowth is turned on, a "keep-initial" option can also be turned on, which limits grass regrowth only to areas that were initially vegetated when the model was initialized. If this is turned off, but regrowth is on, then grass can regrow anywhere.
Finally, an optional seasonality slider can enable seasonal cycling in the growth of grass. The seasonality proportion Ps interacts with a year-length Tyear temporal variable to set up a growing season of Tgrowing ticks and dry season of Tdry ticks:
Tgrowing = Ps \* Tyear
(26)
Tdry = Tyear - Tgrowing
(27)
Note that values of Ps less than 0.5 indicate shorter growing seasons, values of Ps larger than 0.5 indicate longer growing seasons, and Ps = 1 indicates no seasonality. These seasons operate within a yearly cycle of Tyear ticks and overlay grazing regrowth such that grass grazed in the growing season will regrow within that season according the the regrowth time, Tgrass regrowth:
if Tcurrent mod Tyear =\< Tgrowing, nregrown = ngrazed in time step {Tcurrent - Tgrass regrowth}
However, grass that is grazed in the dry season will not regrow again until the start of the next growing season:
if Tcurrent mod Tyear \> Tgrowing, then nregrown = ngrazed in time steps {Tyear - Tdry}
(26)
Thus, at the one extreme, the environmental components of the model can be set as completely static with continual grass coverage on every patch without the effect of grazing, or at the other, very patchy grass that is grazed and then regrows continually or seasonally.
The spatial and temporal scale of the MHPM needs to be set at the start of any modeling experiment. These are determined by setting the values of a core set of temporal and spatial variables. Time steps in the NetLogo interface are called "ticks." The main temporal variable is the number of ticks per year, Tyear , which determines the temporal granularity of the model run. Other temporal variables are set either as proportions of this value or must manually be set in relation to this value. For example, the seasonality proportion variable automatically sets the length in number of ticks of the growing season (and, hence, also the dry season) within a year, but the grazing regrowth timer specifically sets the number of ticks required for a previously grazed patch to regrow during the season of grass growth. Other manual temporal variables are the age of senescence and maximum lifespan of animals, which are each set as numbers of ticks.
The size of the model world is set by determining the number of landscape cells ("patches" in NetLogo) on each axis of the world. The absolute size of a patch in relation to real world distance measures are determined as a function of the number of ticks per year and a ratio of typical distances covered by the modeled prey animals per unit time. Thus, the total number of landscape patches should be great enough to encompass a realistic region of interest that is large enough to allow interesting movement dynamics, but not so large that it unnecessarily increases the computational load required to run the model.
This software is made available under the GNU GPL v3 license. Please cite the authors if you use this model or reuse the code.