Population and Inequality Dynamics in Simple Economies
aa r X i v : . [ phy s i c s . s o c - ph ] J a n Population and Inequality Dynamics in Simple Economies
John Stevenson ∗ December 29, 2020
Abstract
While the use of spatial agent-based and individual-based models has flourished across many scien-tific disciplines, the complexities these models generate are often difficult to manage and quantify. Thisresearch reduces population-driven, spatial modeling of individuals to the simplest configurations andparameters: an equal resource opportunity landscape with equally capable individuals; and asks the ques-tion, ”Will valid complex population and inequality dynamics emerge from this simple economic model?”Two foraging economies are modeled: subsistence and surplus. The resulting, emergent population dy-namics are characterized by their sensitivities to agent and landscape parameters. The various steadyand oscillating regimes of single-species population dynamics are generated by appropriate selection ofmodel growth parameters. These emergent dynamics are shown to be consistent with the equation-based,continuum modeling of single-species populations in biology and ecology. The intrinsic growth rates,carry capacities, and delay parameters of these models are implied for these simple economies. Aggregatemeasures of individual distributions are used to understand the sensitivities to model parameters. Newlocal measures are defined to describe complex behaviors driven by spatial effects, especially extinctions.Inequality measures are defined and applied to surplus economies. Sensitivities of inequality dynamics tomodel parameters are explored. Individual wealth trajectories are examined for insight into the differentdynamics of inequality distributions. This simple economic model is shown to generate significantly com-plex population and inequality dynamics. Model parameters generating the intrinsic growth rate havestrong effects on these dynamics, including large variations in inequality. Significant inequality effects areshown to be caused by birth costs above and beyond their contribution to the intrinsic growth rate. Thehighest levels of inequality are found during the initial non-equilibrium period and are driven by factorsdifferent than those driving steady state inequality.
Keywords: economic simulation, inequality, population dynamics, extinction, agent-basedmodel, individual-based model, population-driven model ”It can scarcely be denied that the supreme goal of all theory is make the irreducible basic elementsas simple and as few as possible without having to surrender the adequate representation of a singledatum of experience.”Albert Einstein (Caprice, 2011)
The modeling of populations of individuals over time on a spatial grid has become a powerful tool in boththe life sciences (DeAngelis and Grimm, 2014; Badham et. al., 2018; Shoukat and Moghadas, 2020), andthe social sciences (Heppenstall, 2020; Chen, 2011). This modelling began in the field of ecology, where it iscalled Individual-Based Modelling, and then expanded into the social sciences, where it is called Agent-BasedModelling. Many of these models, particularly in biology and ecology, are population driven, which exertsselection pressure on the characteristics of populations (Gause, 1934). The ability of these models to supportincreasingly complicated behaviors (both programmed and learned), environments, and sensory inputs hasresulted in investigations that are both broad, across many scientific disciplines (Vincenot 2018), and detailed(di Porcia e Brugnera, 2020; Kohler and Gumerman, 2000). The field has come to recognize a variety of ∗ [email protected] The computational model is described in sufficient detail to ensure reproducibility. Equations for the conser-vation of energy (resources) are developed to both precisely define the computational process and to validatethe conservation of resources. Subsistence and surplus economies are defined, actual and theoretical carrycapacities are calculated, and sensitivities of actual carry capacities to model parameters are developed. Themodel parameters are further simplified and sensitivities of population dynamics to these remaining parame-ters are explored. The population dynamics are put in context with the single-species, equation-based models2f mathematical biology and the intrinsic growth rates, carry capacities, and delay parameters for the simpleeconomic models are estimated.
Table 1 provides the definition of the agents’ characteristics and landscape parameters for generating simpleeconomies. Vision and movement are along rows and columns only. The two dimensional landscape wrapsaround the edges (often likened to a torus). Agents are selected for action in random order each cycle.The selected agent moves to the closest visible cell with the most resources with ties resolved randomly.After movement, the agent harvests and consumes (metabolizes) the required resources. At this point, if theagent’s resources are depleted, the agent is removed from the landscape. Otherwise an agent of sufficient age(puberty) then considers reproduction, requiring sufficient resources (birth cost), a lucky roll of the fertilitydie (infertility), and an empty von Neumann neighbor cell, which are only the four neighboring spaces onestep away by row or column. If a birth occurs in a configuration with zero puberty, the newborn is addedto the list of agents to be processed in this current cycle. Otherwise (puberty ¿0), the newborn is placedin the empty cell and remains inert until the next action cycle. With this approach for the action cycle, noendowments are required whether for new births or for the agent(s) at start-up. Once all the agents havecycled through, the landscape replenishes at the growth rate and the cycle ends.Agent Characteristic Notation Range Units Purposevision v m
1+ resource per cycle consumption of resourcebirth cost bc
0+ resource sunk cost for reproductioninfertility f
1+ 1/probability likelihood of birthpuberty p
0+ cycles age to start reproductionsurplus S
0+ resources storage of resource across cyclesLandscape Characteristic Notation Value Unitsrows – 50 cellscolumns – 50 cellsmax capacity R g R m = 4) and the latter a surplus economy ( m = 3).And, again for simplicity, the vision and movement characteristics are set to equal values of distance.The initial population starts with a single agent with no initial endowment. The population trajectoryover time is dependent on the three growth characteristics, infertility, puberty, and birth cost. The birth costparameter represents the amount of resources the parent agent expends to reproduce. The surplus of theparent agent must be sufficient to provide this amount and it is a sunk cost. The puberty parameter representsthe number of cycles required after birth before the new agent can reproduce. The infertility parameter ( f )defines the probability, P b , that an agent with sufficient resources and space (a free neighboring space) will,in fact, reproduce in this cycle as: P b = 1 /f (1)where f > = 1. This parameter can be seen as a simple way to approximate various more complex reproductivecharacteristics and intrinsic growth rates.Since these parameters will be used to generate many illustrative simple economies, Table 2 provides ashorthand for labeling these parameters for the simple economic model:3imple Economy Parameters symbol labelinfertility f fXXpuberty p pXbirth cost bc bcXXmetabolism m mXsimulation length T XXTable 2: Shorthand Labeling for Simple Economic Models. As an example, f10p1bc40m3 50K is a surpluseconomy ( m of 3 resources per cycle) with f at 10, p at 1 cycle, bc at 40 resources per birth, and simulationlength of 50,000 cycles. The calculation of conservation of energy (resources) confirms the validity of the simulation and provides aprecise description of the computational process. Two energy conservation equations are written, one for thelandscape E L , and one for the population E P . In order to easily write the conservation equations, the energyharvested H ( t ) from the landscape by the agents at time t is defined as: H ( t ) = A ( t − X a =1 r [ c ( a ) , t ] + δ p ∗ B ( t ) X a =1 r [ c ( a ) , t ] (2)where a is the agent index, A ( t −
1) is the list of agents alive at the end of the previous cycle t − B ( t ) isthe list of new agents generated in this cycle t , δ p is one if p equals 0 and 0 otherwise, c ( a ) is the cell locationof an agent indexed as a , and r [ c ( a ) , t ] are the resources in the cell occupied by a that are harvested by a inthe current cycle. The landscape resource conservation equation can now be written as:∆ E L ( t ) = N c X c =1 g c ( t − − H ( t ) (3)where ∆ E L is the change in total resources of the landscape from the previous cycle t − t , N c is the number of cells in the landscape, g c ( t −
1) is the resource added to cell c at the end of theprevious cycle t − g c ( t ) is given as: g c ( t ) = ( g r [ c ( a ) , t ] + g ≤ RR − r [ c ( a ) , t ] r [ c ( a ) , t ] + g > R (4)where R is the maximum resources in a landscape’s cell, and g is the growth rate of resources in landscape’scell. The conservation equation for the resources in the population can now be written as:∆ E P ( t ) = H ( t ) − A ( t ) X a =1 m − D ( t ) X a ( S a ( t ) + m ) − B ( t ) X a =1 [ bc − (! δ p ) m ] (5)where ∆ E P ( t ) is the change in surplus resources stored in the population from the previous cycle to thecurrent cycle, A ( t ) is the list of agents alive at time t , m is the (constant) metabolism, bc is the (constant)birth cost, ! δ p is the Boolean not of δ defined for Equation 2, D ( t ) is the list of agents that died in this cycleand S a ( t ) is the surplus resources of those agents a on list D which have died ( S a ( t ) <
0) in this cycle sothat S a ( t ) + m are the (positive) resources lost upon its death.Equation 5 shows that in addition to the energy consumed by metabolism, there are two additional termsthat remove energy from the system and, therefore, represent sunk costs. The first sunk cost ( P D ( t ) a ( S a ( t ) + m )) represents the stored, surplus resources of those agents that died during this cycle. The second sunkcost ( P B ( t ) a =1 [ bc − (! δ p ) m ]) represents the cost of reproducing the new agents in this cycle. The relationshipbetween these two terms and their relative sizes play an important role in inequality distributions [Section4]. 4able 3 provides example energy balances for representative simple economies. The sum of changes inresources on the landscape (Equation 3) and in the population (Equation 5) are compared to the actual valuesof these resources at the end of the simulations of representative simple economies. These results confirm theabove equations accurately describe the computational process and verify that this process conforms to theconservation of energy (resources).Simple Economy P T ∆ E L P T ∆ E P Landscape Resources at T Population Resources at T f85p1bc0m3 50K 2,603 4,861 2,603 4,861f10p1bc0m3 10K 2,464 646 2,464 646f10p1bc40m3 50K 2,607 16,608 2,607 16,608f1p0bc0m4 6 3,537 0 3,537 0Table 3: Examples of the Energy (Resource) Balance. The energy balances for representative simpleeconomies. The sum of changes in resources on the landscape (Equation 3) and in the population (Equation5) are compared to the actual values of these resources at the end of the simulations. The carry capacity of a landscape is a parameter that self-limits the growth of a single-species population. Ithas both a physical interpretation through the conservation of resources and a modelling interpretation forthe application of equation-based models of populations. The theoretical carry capacity is developed fromthe conservation equations and the actual carry capacities are computed for various parameters of the simpleeconomic model.
The theoretical carry capacity ( K T ) for any simple economy is based on balancing the resources generatedper cycle (∆ E L ) with the resources consumed by the population (∆ E P ) in that cycle assuming no deaths orbirths. Furthermore, the growth of resources assumes that every cell in the landscape is below the maximumcapacity R . Setting Equation 3 equal to Equation 5 with these assumptions and solving for K T yields: K T = N c ¯ m ∗ g (6)where N c is the number of cells in the landscape, g is resource growth per cell per cycle, and ¯ m is the meanmetabolism of the population. The growth rate of resources is uniform across the landscape, providing equalopportunity to all locations. The metabolism, along with vision/movement, is uniform across the entirepopulation, providing equal ability to all agents.A simple subsistence economy has an uniform agent metabolism of four resource units per cycle for eachagent, yielding a theoretical carry capacity of 625 agents. Since the maximum capacity of a landscape cell isalso four resource units, the agents in this economy never have the opportunity to store resources from onecycle to the next. A simple surplus economy, on the other hand, with an uniform agent metabolism of threeunits per cycle for each agent, yields a theoretical carry capacity of 833 agents. This economy does providefor the storage of surplus resources from one cycle to the next and there is no limit to how many resourcescan be stored by an agent. The sensitivity of actual carry capacity to vision/movement with no initial stores is shown in Figure 1. Theactual carry capacity is the mean of the population at steady state. The standard deviation of these meansare also reported in this figure. The population mean and standard deviation are computed in the steadystate window (2,000 through 3,000 cycles for the runs with infertility of 85; and 400 through 1000 cycles forthe others). Determination of when steady state is achieved is discussed in detail in the following sections.5 P opu l a t i on M ean ( agen t s ) K S t anda r d D e v i a t i on o f P opu l a t i on ( agen t s ) a.) Subsistence economy Standard Deviationf10f40f85Populationf10f40f85 5 10 15 20Vision/Movement (cells) P opu l a t i on M ean ( agen t s ) K S t anda r d D e v i a t i on o f P opu l a t i on ( agen t s ) b.) Surplus economy Standard Deviationf10f40f85Populationf10f40f85
Actual Carry Capacities for Subsistence and Surplus Economies
Figure 1: a.) The means and standard deviations of population level for three subsistence (a) and surplus(b) economies by infertility with puberity ( p = 1) and birth cost ( bc = 0) held constant. K represents thecarry capacity of the landscape for each economy and f represents the infertility parameter. The meansand standard deviations are taken over a steady state time window (2,000 through 3,000 cycles for the runswith infertility of 85; and 400 through 1000 cycles for the others). Due to the lack of any surplus resources,the subsistence economies struggle to reach the theoritical carry capacity. The f10p1bc0 parameters for bothsurplus and subsistence economies show the lowest actual carry capacities and an order of magnitude increasein volatility as the vision/movement increases. These increases indicate a transition to an unstable regime.The actual carry capacity falls short of the theoretical capacity of the landscape due to local, spatial effects.These local inefficiencies have greater effect on the subsistence economy due to the agents lack of surplusresources to carry-over from one cycle to the next. All subsistence economies, regardless of infertility, havetrouble approaching the theoretical carry capacity of the landscape with reasonable vision and movement.For both economies, the standard deviations of the means of the populations with infertility 10 (f10) spike upabove the standard deviations of populations with higher infertility. This transition to an order of magnitudehigher variance is characteristic of a change in the dynamic regime of the economy defined by these modelparameters. These regime changes play an important role throughout this research.Based on the relatively flat sensitivity of the carry capacity to vision/movement once the agents cansee and move more than five cells, the vision and movement parameters will be held constant at 6 for thesimple economies of the remaining experiments. Model parameters that remain variable are part of eitherthe reproductive process or defining the simple economic model as subsistence or surplus. Essential to understanding population dynamics generated by this simple economic model are the relation-ships of emergent dynamics to the remaining model parameters. The sensitivities of population dynamicsof both surplus and subsistence economies are computed and discussed for the remaining variable agentcharacteristics of infertility, puberty, and birth cost.
The sensitivities of population dynamics in surplus economies to small and large birth costs holding infertilityat five and puberty at one are given in Figure 2. This figure shows clearly how the birth cost parameteraffects the intrinsic growth rate. These large birth cost economies will play an interesting role in the later6tudies on inequality. The population dynamics at steady state are flat for high birth costs and move throughthe various regimes of constant level; and damped, steady, and chaotic magnitude periodic oscillations forthe lower birth costs. P opu l a t i on ( agen t s ) Birth Cost420 a. Populations with small birth costs for infertitlity 5, puberty 1 P opu l a t i on ( agen t s ) Birth Cost104846444244 b. Populations with large birth costs and infertility 5 and puberty 1
Sensitivity of Population Dynamics to Birth Costs in a Surplus Economies
Figure 2: a.) The population trajectories of surplus economies with various small birth costs (a) and largebirth costs (b) for constant infertility ( f = 5) and puberty ( p = 1). The sensitivity of the trajectories todecreasing birth cost can be seen both in increasing rates of growth to carry capacity, and in the increasingmagnitude of oscillations.The sensitivity of the population dynamics to the infertility parameter ( f ) with puberty and birth costsboth held at zero are given in Figure 3(a). These sensitivities to puberty with infertility held constant at 1and birth cost 0 are given in Figure 3(b). Both figures show the same regimes of constant level; and damped,steady, and chaotic magnitude periodic oscillations as seen for the previous birth cost sensitivities. Thesesensitivities also give rise to the first examples of a population going extinct. Individual-based models providesignificant and unique insights into extinctions. 7
50 100 150 200 250 300Time (cycles) P opu l a t i on ( agen t s ) Infertility21111 a. Populations with varied infertilities with birth cost and puberty 0. P opu l a t i on ( agen t s ) Puberty30150 b. Populations with varied puberty for birth cost 0 and infertility 1
Sensitivities of Population Dynamics to Infertility and Puberty in Surplus Economies
Figure 3: a) The population trajectories of surplus economies with varied infertility for constant puberty( p = 0) and birth cost ( bc = 0). b) The population trajectories of surplus economies with varied puberty forconstant infertility ( f = 1) and birth cost ( bc = 0). Decreasing values of infertility or puberty increase thegrowth rate towards carry capaciity and increase the magnitude of oscillations. Both these graphs show thatat high enough growth rates, chaotic extinctions occur [Section 3.2].While population trajectories attain their steady state modes quite quickly; the individual, local charac-teristics of the population, such as age or surplus resources, may not. In fact, these individual characteristicsmay take as long as another order of magnitude of time to reach steady state. These transition periodsturn out to be very important for many individual level measurements of population characteristics and aretreated in detail in Section 4.0. Population growth in a subsistence economy is a more challenging activity for the individuals. By definition,the amount of resources consumed during a cycle is equal to the maximum available from a landscape cell.There are no resources remaining to support replication unless the birth cost is zero. Figure 4 below showsthe sensitivity of population growth to infertility and puberty. All the now familiar regimes are representedhere as well. 8
50 100 150 200 250 300Time (cycles) P opu l a t i on ( agen t s ) Infertility21111 a. Populations with varied infertility for birth cost and puberty 0 P opu l a t i on ( agen t s ) Puberty31161 b. Populations with varied puberty for birth cost 0 and infertility 1
Sensitivities of Population Dynamics to Infertility and Puberty in a Subsistence Economies
Figure 4: a) The population trajectories of subsistence economies with varied infertility for constant puberty( p = 0) and birth cost ( bc = 0). b) The population trajectories of subsistence economies with varied pubertyfor constant infertility ( f = 1) and birth cost ( bc = 0). These trajectories show similiar sensitivities as thesurplus economies (Figure 8) but with a lower actual carry capacity and somewhat more chaotic magnitudesof the oscillations. Chaotic extinctions [Section 3.2] are present in both graphs and the f1p0bc0 extinction isthe first example of a density-limiting high growth extinction [Section 3.1].For the subsistence economies, extinction events are more common. Figure 4(a) shows an immediateextinction due to a very high intrinsic growth rate. Figures 4(b) and 3(a) provide examples of extinctionsresulting from chaotic magnitude periodic oscillations. The fields of mathematical biology and ecology developed equation-based continuum modeling of single speciespopulations, models both continuous and discrete (Murray, 2002; Kot, 2001). A continuous homogeneousmodel of single species population N ( t ) was proposed by Verhulst (1838): dN ( t ) dt = rN (1 − NK ) (7)where K is the steady state carry capacity, t is time, and r is the intrinsic rate of growth. This model repre-sents self-limiting, logistic growth of the population. This macroscopic model of a continuous, homogeneouspopulation is quite descriptive and allows the exact solution N ( t ) = K [1 + ( KN − e − rt ] (8)where N is the initial population.Table 4 provides the implied intrinsic growth rates r i and implied carry capacity K i based on fitting thesolution to the Verhulst Model (Equation 8) to the simple surplus and subsistence economies generated bythe simple economic model with representative infertility, puberty and birth cost parameters.Figures 5 and 6(a) show how the Verhulst Model with these implied parameters compare to the actualpopulation trajectories of these simple economies. 9conomic Model Implied Growth Rate r i Implied Carry Capacity K i Plotf85p1bc0m3 0.026 843 Figure 5(a)f10p1bc40m3 0.033 825 Figure 5(a)f20p1bc0m4 0.094 585 Figure 5(a)f10p1bc0m4 0.179 589 Figure 6(a)f10p1bc0m3 0.242 860 Figure 5(b)f5p1bc0m4 0.335 571 Figure 6(a)f5p1bc0m3 0.470 870 Figure 5(b)Table 4: Simple Economic Models fit to the Verhulst Model (Equation 7). P opu l a t i on ( agen t s ) a. Implied parameters in stable regimes f85p1bc0m3actualfitted model f20p1bc0m4actualfitted modelf10p1bc40m3actualfitted model 0 50 100 150 200Time (cycles) P opu l a t i on ( agen t s ) b. Implied parameters in oscillating/chaotic regimes f10p1bc0m3actualfitted modelf5p1bc0m3actualfitted model Implied Growth Parameters from the Continuous Verhulst Model
Figure 5: Implied growth parameters for (a) representative stable, simple economies (f85p1bc0m3,f20p1bc0m4 and f10p1bc40m3) and for (b) representative oscillating/chaotic, simple surplus economies(f10p1bc0m3 and f5p1bc0m3). These implied parameters are least-squares fits to the continuous VerhulstModel (Equation 7) of population trajectories generated by the simple economic model. While these param-eters fit the intrinsic growth rate of Verhulst Model and provide insights into the relationship of the model’sgrowth parameters to the intrinsic growth rate, the Verhulst model does not admit oscillating behaviors asseen with the simple economic model.As can be seen from these figures, while the Verhulst Model fits the initial phase of growth well, it doesnot model the oscillating population levels at the higher rates of intrinsic growth. The types of periodicand chaotic oscillations seen in these simple economies are often generated by population models that arediscrete, that contain time delays, or both (Liz, 2014)10
50 100 150 200Time (cycles) P opu l a t i on ( agen t s ) a. Implied parameters in oscillating/chaotic Regimes f10p1bc0m4actualfitted modelf5p1bc0m4actualfitted model 0 20 40 60 80Time (cycles) P opu l a t i on ( agen t s ) Intrinsic Growth Rate0.0250.47 22.7 a. Discrete Verhulst process regimes
Implied Subsistence Growth Parameters and Discrete Verholst Process Regimes
Figure 6: a) Implied growth parameters for representative oscillating/chaotic, simple subsistence economies(f10p1bc0m4 and f5p1bc0m4). These implied parameters are least-squares fits to the continuous VerhulstModel (Equation 7) of population trajectories generated by the simple economic model. The Verhulst Modelwith the implied growth parameters from the simple econommic model matches the growth rate well but isnot capable of modelling the oscialltions generated by the simple economic model. b) Representative criticalregimes of the discrete Verhulst process (Equation 10). The discrete Verhulst process, while representing thediscrete version of the continuous Verhulst Model, is capable of generating oscillations with decaying, steady,and chaotic magnitudes, similar to what emerges from the simple economic model.Researchers in the fields of biology and ecology have used these discrete and delayed population modelsto handle, for example, species that have no overlap between generations (Murray, 2002) or have specificbreeding seasons (Kot, 2001). Writing the Verhulst model (Equation 7) as a difference equation: N ( t + 1) − N ( t ) = rN ( t )[1 − N ( t ) /K ] (9)and rearranging yields the discrete logistic growth equation, often referred to as a discrete Verhulst process(May, 1974): N ( t + 1) = [1 + r − rN ( t ) K ] N ( t ) (10)where K is the carry capacity treated as free parameter. The discrete Verhulst population trajectories havedifferent regimes that are quite similar to the trajectories of simple economies and very different from thecontinuous model. Figure 6(b) plots the various dynamic regimes generated by critical intrinsic growth ratesfor the discrete Verhulst process: constant level; and damped, steady, and chaotic magnitude periodic oscilla-tions. The intrinsic growth rates representative of each of the regimes in the Verhulst process (Equation 10)are much greater than the implied growth rates (Table 4) from the continuous Verhulst model (Equation 7).Since the aggregate behavior of the simple economic model and the implied growth rates were in agreement,the discrete Verhulst process may still be missing a significant factor.Once a landscape cell’s resources have been consumed, for a growth rate of one resource per cycle, it takesfour cycles for a complete restoration of the cell’s resources. A subsistence economy requires four cycles ofgrowth to match the agent’s metabolism while surplus economies require only three cycles. To account forsimilar delays in animal populations, Hutchinson (1948) modified the Verhulst process (Equation 10) as: dN ( t ) dt = rN ( t )(1 − N ( t − τ ) K ) (11)11y incorporating an explicit time delay τ in the self-limiting term. The resulting discrete-delayed logisticequation (Wright, 1955), often referred to as the Hutchinson-Wright equation, (Kot, 2001) is then N ( t + 1) = [1 + r − N ( t − τ ) K ] N ( t ) (12)Figure 7(a) plots the population dynamics generated by this Hutchinson-Wright equation with τ equalto three and the implied growth rates and carry capacities from the continuous Verhulst equation for thesimple surplus economies (Table 4). Figure 7(b) plots the same results for τ equal to four and the impliedparameters for the simple subsistence economies (Table 4). Magnitudes of intrinsic growth rates for eachof the dynamic regimes generated by the Hutchinson-Wright process show better agreement with impliedgrowth rates for the simple economies by matching the delay with replenishment times. P opu l a t i on ( agen t s ) Implied Growth RateImplied Growth Rate0.0260.033 0.2420.47 b. Implied rates from surplus economies using a delay of 3 cycles P opu l a t i on ( agen t s ) Implied Growth Rate0.0940.179 0.335 b. Implied rates from subsistence economies using a delay of 4 cycles
Discrete Hutchinson−Wright Processes
Figure 7: Implied intrinsic growth parameters for representative simple economies (Table 4) applied to thedescrite Hutchinson-Wright process (Equation 12). a) Surplus economies applied with a delay parameter ( τ )of 3 cycles and b) subsistence economies applied with a delay parameter of 4 cycles. The critical intrinsicgrowth rates and delay parameters of the Hutchinson-Wright process show good agreement with the intrinsicgrowth rates implied from fitting the population trajectores generated by the simple economic model tothe continuous Verhulst model (Equation 7) when using the appropiate delay parameter for surplus andsubsistence landscape growth rates. The population growth of simple economic models have generated distinct dynamic regimes: initial exponen-tial population growth; constant level; damped, steady, and chaotic magnitude periodic oscillating populationtrajectories; and extinction events. A comparison of these dynamics with the family of single species logisticequations validates these computational results to well established global population models of biology andecology and provides insight into how parameters of the simple economic model map to intrinsic growth ratesof the equation-based models of mathematical biology.
Studies of extinction events are aided by the spatial-temporal resolution provided by individual-based models.In particular, for these simple economies, extinction events illustrate both the importance of a precise under-standing of the simulation loop as well as the benefits of spatial resolution of individuals. This section will12ook at both immediate extinctions due to very high intrinsic growth rates and delayed extinctions resultingfrom regimes with chaotic trajectories. As part of this discussion, measures of self-limiting growth and localversus global resource availability are developed.
As the previous sensitivity studies have shown, certain combinations of model parameters result in extinctions.With parameters set to maximum intrinsic growth in a subsistence economy (f1p0bc0m4), the populationramps up quickly and within five cycles goes extinct for all one hundred separate runs. Figure 8(a) plots thepopulation trajectories of these hundred separate runs. P opu l a t i on ( agen t s ) Population trajectories for 100 runs R e s ou r c e s P opu l a t i on ( agen t s ) Resources available for a representative run
Resourceslandscaperequired Agentsalivedead
High Intrinsic Growth Extinction in a Subsistence Economy
Figure 8: a) The population trajectories for 100 runs of a high intrinsic growth simple subsistence economy(f1p0bcom4). All the runs are extinct by the fifth cycle and the initial growth is density limited for the firsttwo cycles. b) The resources available versus resources required by the current population for a sample runtaken from the 100 runs. The available landscape resources are insufficent to support the population in cycle3 and extinction is predicted by these global measures.Figure 8(b) shows the depletion of the resources within the landscape below that required by the currentpopulation for one of these runs at cycle 3. The last few agents alive in cycle 4, even after surviving theresulting large die off, expire due to an apparent lack of local resources though the global resources would nowsupport the remaining population. This phenomenon of the local landscape being insufficient to support apopulation even though the total landscape resources are sufficient is common in most observed extinctions.A measure of these locally available resources is developed in Section 3.2 on chaotic extinctions. First, thedensity limitation of initial growth of this simple economy is examined.Starting with one agent and an initial landscape maxed out at four resources in every cell, there are manybirths during the very first cycle. With puberty set to 0 cycles, each new agent can immediately reproducein the cycle in which it was born. What prevents the landscape from completely filling immediately is therequirement for an agent to reproduce that one of the four neighboring cells of that agent be unoccupied.Following the agents’ movement for resources and then reproduction, it can be seen in Figure 9(a) how onetrajectory hits the no-empty-neighbor condition after 30 births during the first cycle. Figure 8(a) shows thatall the one hundred trajectories hit this local density-limited growth within 200 births. With such fecundity,even with this density-limited growth, it is not surprising that the population exceeds the theoretical carrycapacity within three cycles and goes extinct within five cycles for all the runs. A measure of this local13ensity limitation, a crowding factor C fj , is defined for the reproducing agent in c j as C fj = V j (1) X i =1 c emptyi (13)where V j (1) are the von Neumann neighbors at a distance of 1 from cell c j , and c emptyi is a Boolean functionwith value zero if c i is occupied by another agent, otherwise it equals one, yielding a count of the number ofempty, neighboring cells. Figure 9(b) plots the crowding factor for each reproducing agent from Figure 9(a),demonstrating that the lack of a empty von Neumann neighbor ended the first cycle. columns r o w s
21 22 23 24 25 26
StartEnd
Sequence of new agent sites agent c r o w d i ng Crowding factor by agent
Local Denisty−Limited Growth During the First Cycle for a High Growth Subsistence Economy
Figure 9: a) New agent birth sites during the first cycle of high growth for a simple subsistence economy(f1p0bc0m4). The puberty value of zero allows a new agent to immediately enter the action queue. Thesequence ends when there are no empty sites ( V j (1)) available for the latest agent in cell j . b) The crowdingfactor (Equation 13) for the sequence of new agents shown in (a). When the crowding factor reaches zero,the growth is density-lmited and the cycle ends. Setting the puberty to one cycle significantly reduces the intrinsic growth rate and delays but does notnecessary avoid extinction. Figure 10(a) presents a histogram of the extinction time of all thousand runs ofsuch a simple economy (f1p1bc0m4), with only a few outlying runs extending past 800 cycles. Figure 10(b)plots the chaotic population trajectory for the run that lasted the longest in this set of 1000.14
200 400 600 800 1000Extinction Time (cycles) F r equen cy a) Extinction times for f1p1bc0m4 economies P opu l a t i on ( agen t s ) b) Sample population trajectory for a f1p1bc0m4 economy Chaotic Extinction in a Subsistence Economy
Figure 10: a) Histogram of extinction times for a 1,000 runs of a chaotic subsistenace economy (f1p1bc0m4).Most extinctions occur within the first few hundred cycles, with the longest lasting run reaching over 1,000cycles. b) The population trajectory for the longest lasting run showing the oscillations of chaotic magnitude.All the trajectories in (a) display this chaotic behavior resulting in stochastic extinction times for the samesimple economy.Of these 1000 f1p1bc0m4 trajectories, a trajectory with an extinction time of 162 cycles is selected formore detailed examination of this different mechanism of extinction than seen in Section 3.1. The globalor macroscopic measures of population and landscape resources of this selected trajectory are given in theFigure 11(a). P opu l a t i on ( agen t s ) R e s ou r c e s PopulationGlobal Resources a. Population trajectory and total landscape resources P opu l a t i on ( agen t s ) b. Phase space representation of the extinction trajectory Global Measures of an Extinction of a Subsistence Economy R lj ( t ) to be the local measure of resources within range of an agent occupying c j as R lj ( t ) = V j ( v ) X i =1 r ( c i , t ) (14)where V j ( v ) are all the von Neumann neighbors within movement/vision range v of cell c j , and r ( c i , t ) arethe currently available resources in c i at time t . Averaging over all the agents A ( t ) defines an averaged localmeasure of resources available to the population ¯ R ltotal as¯ R ltotal ( t ) = A ( t ) X i =1 R lj ( t ) /A ( t ) (15)Figure 12(a) provides ¯ R ltotal and the global population versus time. ¯ R ltotal is a predictor of a pending extinctionin contrast to the total resources available [shown in Figure 11(a)] which does not recognize an impendingextinction event. P opu l a t i on ( agen t s ) sc ape R e s ou r c e s PopulationLocal Resources a. Population and locally available landscape resources
140 145 150 155 160Time (cycles) P opu l a t i on ( agen t s ) sc ape R e s ou r c e s Local ResourcesPopulation b. Detail of final oscillations with locally available resources
Local Measures for an Extinction Event
Figure 12: a) The population trajectory and averaged locally available landscape resources ( ¯ R ltotal ) for asimple subsistence economy (f1p1bc0m4). b) Detail of the final two oscillations leading to extinction. Forthe last three cycles, the averaged locally available resources were never enough to support the dwindlingpopulation though the global resources, shown in Figure 11, had fully recoverd. The last remaining agentsperish in a local desert with plentiful resources too far away.Figure 12(b) provides greater detail of the final oscillations that results in the extinction event. Spatial-temporal characteristics of this extinction event are examined in further detail in the next section.16 .3 Microstructure of Extinctions To help understand the details of how an extinction event is driven by non-equilibrium, local, spatial, andstochastic effects, Figure 13 provides a spatial visualization for four cycles of the agent locations and theresources available for the last two minimums of the run detailed in Figures 11 and 12. Figure 13(a) and (b)show the penultimate oscillation and recovery. a.) cycle 144 Legend: Cell Resource: 0 (white) to 4 (dark gray), Agent occupying cell (black)b.) cycle 149c.) cycle 158 d.) cycle 161
Microstructure of an Extinction Event
Figure 13: a) When the penultimate wave of agents pass through the rich landscape, a couple agents manageto survive behind the wave. b) As the last agents of the wave perish in a local desert, the next generationblooms. c) For this final wave, no agents survive behind the wave as it passes. d) The last cycle with theremaining agents all in a local desert and about to perish. This extinction event is driven by non-equilibriummovement dynamics, and spatially local resource densities. Together these factors produce a highly stochasticprocess leading to certain extinction but at an unpredictable time.At the agent level, the agents form a cylindrical wave of resource depletion moving from a starting smallgroup of agents at the center and ranging radially outward until all the resources have been consumed. Note17hat two agents in Figure 13(a) have survived in the center after the consuming wave moves out radially.Figure 13(b) shows the last remnants of that wave expiring at the edges as the next generation blooms fromthe agents that survived in the center. Thus the population as a whole survived this oscillation. Figure13(c) shows the next oscillation, with the wave of agents flowing out from the center again but this time,for the first time, no agents in the center are able to survive the passing of the wave. Thus, Figure 13(d)shows the extinction of the population as the last remnants of the wave perish in a local desert with noagents left behind in the middle to seed the next generation. This extinction is purely a local, and spatialphenomenon since the total resources are as much or more than the numerous previous oscillations whichavoided extinction. The extinction is also stochastic since the survival of a few agents in the center is likelybut not certain.
In addition to the spatial resolution that is so important in understanding extinction events, additionalmeasures of distributions of individual characteristics are available with this simple economic model. Threeof these measures are the individuals’ ages, surplus resources (wealth), and the deaths per cycle. Thissection will first investigate the sensitivities of the populations’ mean age and inequality distributions to themodel’s parameters at steady state. Four representative surplus economies are then identified as models forfurther study into the individuals’ distributions. Large differences in inequality due to the different growthparameters are highlighted and the causes of inequality in homogeneous (equal ability) agent populations in anhomogeneous (equal opportunity) resource environment are considered. Emphasis is placed on the differencesin these characteristics between steady state (equilibrium) and transient (non-equilibrium) conditions.
A proxy for measurement of inequality in populations often used and misused is the Gini Coefficient, G ,defined as: G = E ( | S ′ − S ” | ) / (2 µ ) (16)where E is the expectation operator, S ′ and S ” are the surplus resources of different agents and µ is the meanof the surplus resources µ = E ( S ) (Yitzhaki and Schechtmann, 2012).Computationally, Equation 16 was implemented as: G = P Ai =1 P Aj =1 | S i − S j | Aµ (17)where i and j are indices to run through the entire population A of agents who have S index surplus resources.Though there are many problems with describing whole distributions with a single number, by computing theGini Coefficient over the entire population rather than sampling the population, by not assuming a particulardistribution, and by only using Gini Coefficients as relative measures for comparing populations of the samesize, these inaccuracies can be minimized (Fontanari et. al., 2018). The Gini Coefficient is particularlysuspect when used for comparisons of populations that are undergoing large oscillations in size over time(Talib, 2015).In order to select appropriate model economies that will generate various inequality distributions, sen-sitivities of the mean age and inequality (as imperfectly expressed by the Gini Coefficient) to the growthparameters are computed. Care must be exercised to ensure the specific population has reached a steadystate for all its measurement parameters, not just population size. While the population will hit its actualcarry capacity quite quickly, the mean age of the population can take from one hundred to one hundredthousand cycles to reach its steady state.Figure 14(a) plots the sensitivity of these distributional measurements to birth cost, and Figure 14(b),the sensitivity to puberty. The large variance in Gini Coefficient for repeated runs in the same configurationunderscores the imperfect value of this metric at a specific cycle in oscillating and/or chaotic regimes.18
20 40 60 80Birth Cost . . . . G i n i C oe ff i c i en t GiniRun 1Run 2 M ean A ge o f P opu l a t i on Mean AgeRun 1Run 2 a. Birth cost sensitvity . . . G i n i C oe ff i c e i n t GiniRun 1Run 2 M ean A ge o f P opu l a t i on Mean AgeRun 1Run 2 b. Puberty sensitivity
Inequality and Mean Age Sensitivities for Surplus Economies at Steady State
Figure 14: a) The sensititivty of the mean age and Gini Coefficient to birth cost with constant infertility( f = 10) and puberty ( p = 1) at the end of the 130K to 150K cycles steady state window. The addition ofbirth costs to this simple f10p1 economy immediately and significantly reduces the inequality and holds itat these low levels as the birth costs continue to increase. The effeccts of increasing birth costs on the meanage of the population, in contrast, are quite linear. b) The sensititivty of the mean age and Gini Coefficeintto puberty with birth cost ( bc = 0) and infertility ( f = 10) at the end of the 10k to 20K cycles steady statewindow. Increasing puberty reduces inequality and increases mean age for the f10bc0 simple economy thoughthe Gini Coeficient measurements have a large variance, suggesting a chaotic regime.Figure 15(a) plots the sensitivities of inequality and mean age to infertility. The variance of the GiniCoefficient at small infertility will be addressed below. Figure 15(b) presents the relationships of mean agesand Gini coefficients to the mean of the deaths per cycle ( ¯ d c ) over the steady state window for variationsof the agents’ growth characteristics. In Figure 15(b) it becomes clear that the slope of the relationship ofmean age to mean deaths per cycle is significantly different for birth cost sensitivity than it is for puberty andinfertility, This difference in slopes is the first indication that the individual dynamics of surplus economieswith significant birth cost are substantially different than without birth cost. The sensitivity of the GiniCoefficient to these parameters is more complex. The change in slope of mean age for increasing deathsper cycle and the double slope reversal for the inequality measures indicate a phase change in the dynamicsof these distributions. In general, across all these sensitivities, the higher the mean deaths per cycle, thelower the mean age and the higher the inequality. These tendencies are based solely on economies that havereached steady state and do not apply to non-equilibrium economies still in transition.19
20 40 60 80 100Infertility . . . . . G i n i C oe ff i c i en t GiniRun 1Run 2 M ean A ge o f P opu l a t i on ( cyc l e s ) Mean AgeRun 1Run 2 a. Mean age and Gini sensitivity to Infertility . . . . . G i n i C oe ff i c en t Giniinfertility M ean A ge o f P opu l a t i on ( cyc l e s ) Mean Agepubertyinfertilitybirth cost predicted mean age b. Mean age and Gini versus mean deaths per cycle
Inequality and Mean Age Relationships for Surplus Economies at Steady State
Figure 15: a) The sensititivty of mean age and Gini Coefficent measurements to infertility with constantpuberty ( p = 1) and birth cost ( bc = 0) at the end of the 40K to 50K cycles steady state window. b) Therelationship of mean age and Gini Coefficent to mean deaths per cycle over a broad range of model parameters(using the same steady state windows as before). A prediction of mean age based on probability of death percycle (Equation 18) is plotted.The predicted mean age model shown in Figure 15(b) is based on a Monte Carlo simulation of the steadystate population A run over time using the probability of an (mean) agent’s death in any one cycle P ¯ d c tobe given as P ¯ d c = ¯ d c /A (18)This prediction shows good agreement with the mean age data for the birth cost sensitivity but gets theslope wrong for the puberty and infertility sensitivities and gives no indication of the change in linearity ofthe slope at high ¯ d c . The mean deaths per cycle metric, however, gives a good intuitive feel for what ishappening to the population (level of carnage) with a particular configuration of growth parameters.Figure 16(a) plots the relationship of the population’s total wealth to mean age. The sensitivities of allthree model growth parameters show increasing total wealth with increasing mean age, with similar andnearly constant slopes. The total wealth increases with increasing mean age even for increasing birth costs,which represent significant sunk costs. Figure 16(b) presents the relationship of the Gini Coefficient tothe mean age of the population. A couple of complexities are apparent in Figure 16(b). First, two verydifferent populations emerge with similar mean ages but significantly different inequality measures. Second,as mean age decreases to and below 10, an apparent phase transition occurs for both the total wealth andthe inequality with two slope reversals. 20
10 100 1000 10000Mean Age of Population (cycles) + + + + + + + + T o t a l W ea l t h o f P opu l a t i on (r e s ou r c e s ) Detail providedIncreasing Birth CostIncreasing Puberty Increasing Infertilitypubertyinfertilitybirth cost a. Total wealth versus mean age at steady state . . . . G i n i C oe ff i c i en t Detail providedIncreasing Birth CostIncreasing PubertyIncreasing Infertilitypubertyinfertilitybirth cost b. Gini versus mean age at steady state
Total Wealth and Inequality Relationships to Mean Age for Surplus Economies
Figure 16: a) The relationship of population’s total wealth to mean age across a broad range of modelparameters at steady state. The Details on the boxed area are provided by Figure 17(a). b.) The relationshipof the Gini Coeficient to mean age across a broad range of model parameters at steady state. Details on theboxed area are provided by Figure 17(b). Increasing birth costs increase total wealth of the population in asimilar fashion to other model parameters but have a strikingly stronger effect on inequality.Figure 17(a) provides detail of the measurement of total wealth and mean age and their one standarddeviations from the highlighted (boxed) area in Figure 16(a). Figure 17(b) provides the detail of the mea-surements of the Gini Coefficient and mean ages and their one standard deviations from the highlighted(boxed) area in Figure 16(b). There is a significant increase in variance and a loss of one-to-one mapping ofGini Coefficient to mean age as the infertility passes through 10. This regime change is consistent with theHutchinson-Wright process (Equation 12) with a delay of 3 cycles and an implied rate of growth increasingthrough 0.242 as discussed in Section 2.5 and plotted in Figure 7(a). The means and standard deviations forFigure 17 were calculated over a steady state window of time from 10,000 to 20,000 cycles for mean ages lessthan 21 and from 200,000 to 225,000 cycles for mean ages greater than 20.21 T o t a l W ea l t h o f P opu l a t i on (r e s ou r c e s ) pubertyinfertilitybirth cost a. Detail of mean age versus total wealth at steady state . . . . . . . G i n i C oe ff i c i en t pubertyinfertilitybirth cost a. Detail of Gini versus mean age at staedy state Total Wealth and Inequality Realtionships to Mean Age (Detailed Views)
Figure 17: a) The means and standard deviations of the total wealth and mean ages around the transitionto high variance. The loss of one-to-one mapping of total wealth to mean age and the increase in varianceindicate a transition to chaotic magnitude oscillations. b) The means and standard deviations of the GiniCoeficients and mean ages around the transition to high variance. The Gini Coefficeint sensitivity to meanage mirrors the transition behavior of the total wealth.
From the sensitivities generated in the previous sections, three specific surplus economies with four populationdistributions are identified as model economies for more detailed study. Table 5 defines and labels thesedistributions taken from the model surplus economies.Name Fertility Birth Cost Growth Rate Gini Mean Age Total Wealth Steady State(label) (1 /P b ) (resources) ( cycles − ) (-) (cycles) (resources) (cycles)f85p1bc0m3 50K 85 0 0.026 0.530 443 4861 35,000f85p1bc0m3 6K 85 0 0.026 0.777 1327 17,132 6,000*f10p1bc0m3 50K 10 0 0.242 0.669 12.6 509 700f10p1bc40m3 50K 10 1 0.332 0.332 2261 16,608 25,000Table 5: Definitions of Model Economies. (* not at steady state)The trajectories of mean age of whole and elite populations for two model economies (f85p1bc0m3 andf10p1bc40m3) are given in Figure 18(a). The elite population consists of those agents whose wealth ranks inthe top 10% of the population. The f85p1bc0m3 economy has a very long relaxation time of 35,000 cyclesto reach steady state. Of particular interest is the large peak of mean age of over 8,000 cycles for the elitepopulation at a time of 10,000 cycles, dropping at a relative constant rate to a mean age of under 1,800 cyclesat steady state. The whole population has a maximum mean age of 1,327 cycles at about 6,000 cycles. Thispopulation distribution at 6,000 cycles is the fourth model population (f85p1bc0m3 6K) to be examined indetail though it is not at steady state. The second simple economy shown in Figure 18(a) is f10bc40m3 50Kwith a relaxation time of 25,000 cycles. This model economy displays a moderate relaxation time of 10,000cycles to steady state with the highest steady state mean age of over 2,000 cycles. Figure 18(b) presents thesemeasurements for the remaining model economy (f10p1bc0m3 50K). This model economy has the shortestrelaxation time of 700 cycles and mean ages under 20 cycles.22 M ean A ge ( cyc l e s ) top 10% f85p1bc0mean f85p1bc0top 10% f10p1bc40mean f10p1bc40 a.) Elite and total mean ages for f85p1bc0m3 and f10p1bc40m3 M ean A ge ( cyc l e s ) top 10%mean b.) Elite and total mean Ages for f10p1bc0m3 Mean Age Relaxation Times for Model Surplus Economies
Figure 18: a) Elite (top 10%) and overall mean ages for surplus economies f85p1bc0 and f10p1bc40. b) Elite(top 10%) and overall mean ages for surplus economy f10p1bc0. The large inequalities are generated duringthe population’s growth to carry capacity and persist for periods much longer than the time to reach carrycapacity. Model economy f85p1bc0m takes the longest time to attain steady state at over 30,000 cyles.For over a hundred years, the Lorenz Curve has provided both a visual and mathematical means tocompare the inequality of two different wealth distributions. The curve’s construction was defined by Lorenz(1905): ”Plot along one axis cumulated (sic) per cents. of the population from poorest to richest, andalong the other the per cent. the total wealth held by these per cents. of the population.”
Figure 19(a) presents the Lorenz Curve for these four distributions at the end of the simulation as shown inTable 5. The model economy with highest steady state inequality is f10p1bc0m3 which has a limited numberof wealth buckets, the highest turnover of agents, and the lowest mean age. Figure 19(b) presents a histogramof the individual surplus resources (wealth) for the model economies f85p1bc0m3 and f10p1bc40m3 at steadystate. This figure gives a good representation of inequality by highlighting the differences between a highlyunequal wealth distribution (f85p1bc0m3) and a much more equal distribution (f10p1bc40m3).23 .0 0.2 0.4 0.6 0.8 1.0Fraction of Population . . . . . . C u m u l a t i v e F r a c t i on o f W ea l t h Equality, Gini:0f10 p1 bc40 at 50K, Gini:0.33f85 p1 bc0 at 50K, Gini:0.53f10 p1 bc0 at 50K, Gini:0.65f85 p1 bc0 at 6K, Gini:0.78 a. Lorenz Graph of the four model economies F r equen cy + ( agen t s ) b.) Actual wealth Distributions at steady state f85 p1 bc0, Gini:0.530f10 p1 bc40, Gini:0.332 Inequality Meausures for the Model Surplus Economies
Figure 19: a) The Lorenz Curve of the agents’ surpluses for the four model economies. b.) Histograms ofthe agents’ surplus resources for the f85p1bc0 and f10p1bc40 surplus economies at steady state (50K cycles).The number of agents at a level of surplus resources is one less than shown on the graph (frequency +1) toenable logorithmic scaling of the vertical axis. The large difference in wealth distribution between these twosimple economies is evident.Figure 20(a) provides the wealth distribution and cumulative wealth fraction for the most unequal modeleconomy (f85p1bc0m3 6K). This distribution, though having attained a steady population, is still far fromequilibrium for mean age and wealth distribution. Its inequality is the highest, its total wealth is thehighest, and its wealthiest bucket is a record 400 resources. Figure 20(b) presents the wealth distributionand cumulative wealth fraction for model economy f10p1bc0m3 50K, which is the most unequal economy atsteady state among the four model economies, has the lowest mean age for the population and, as shall beseen, has the highest death rates. The poverty of this economy is highlighted by the maximum wealth bucketof only five resources. 24
100 200 300 400Stores at 6K cycles (maximum inequality) F r equen cy ( agen t s ) . . . . . . C u mm u l a t i v e F r a c t i on o f W ea l t h a.) Surplus economy f85p1bc0 at 6K cycles, Gini: 0.777 F r equen cy ( agen t s ) . . . . . . C u mm u l a t i v e F r a c t i on o f W ea l t h b.) Surplus economy f10p1bc0 at 10K cycles, Gini: 0.651 Actual Wealth Distributions for Model Surplus Economies
Figure 20: a) Histogram and cummulative fraction of agents’ surplus resources for f85p1bc0 at maximuminequality (6K cycles). This distribution is far from equilibirum and represents the greatest inequality seen inthe model economies. b) Histogram and cummulative fraction of the agents’ surplus resources for f10p1bc0at steady state (50K cycles). This distribution has the smallest wealth bucket, with no agents storing morethe five resources.Figure 21 presents density histograms of deaths per cycle for the three steady state model economies.These densities are taken over steady state windows of 35,000 to 50,000 cycles for f10p1bc40m3 and f85p1bc0m3,and 600 through 3,000 cycles for f10p1bc0m3. This local measure, based on the individual deaths, also givesa good approximation of the birth rate of these economies since, at steady state, the births per cycle mustequal the deaths per cycle over a short time. D en s i t y ( % ) a.) Model economy f10p1bc40 mean D en s i t y ( % ) mean b.) Model economy f85p1bc0
50 100 150Deaths per Cycle D en s i t y ( % ) mean c.) Model economy f10p1bc0 Frequency of Deaths Each Cycle for the Model Surplus Economies
As all these surplus model economies show, the initial agents (founders), born into a underpopulated and richlandscape, have a tremendous advantage in building up their personal wealth before the population reachesthe actual carrying capacity. Figure 22 gives the initial wealth histories for the early founders of each of thethree model economies. A gen t S t o r e s (r e s ou r c e ) a.) Founding agents’ wealth for f85p1bc0 Agent IDAgent 0Agents 1−29 A gen t S t o r e s (r e s ou r c e ) b.) Founding agents’ wealth for f10p1bc40 Agent IDAgent 0Agents 1−11 A gen t S t o r e s (r e s ou r c e ) c.) Founding agents’ wealth for f10p1bc0 Agent IDAgent 0Agents 1−39
History of the Founding Agents’ Wealth for the Model Surplus Economies
Figure 22: a) a.) Wealth history for the first thirty agents of the f85p1bc0m3 economy. b.) Wealth historyfor the first twelve agents of the f10p11bc40ms economy. c) Wealth history for the first forty agents ofthe f10p1bc0m3 economy. The individual wealth histories of the founding agents of each model populationreveal both the outsized surpluses (relative to steady state) stored by these founders as well as how long thefounders’ wealth persists. The wealth history of the founders of the model economy with non-zero birth cost(b) is of a much different character than the wealth histories of the founders for the other model economies(a) and (c).As Figure 18 of relaxation times previously showed, mean age and inequality of these populations do notattain steady state until these founding agents have given up their surpluses and expired, the last foundertaking 35,000 cycles. The most unequal population was attained by f85p1bc0m3 at 6,000 cycles, which had amean age of 1,327 and a total wealth of 17,132 resources [Figure 22(a)]. Even with such elite wealth and longlifespans, the peak total wealth of this economy not at equilibrium is only comparable to the steady statetotal wealth (16,608 cycles) and mean age (2,261) of the most equal model economy f10p1bc40m3 at steadystate[Figure 22(b)]. The relationships of inequality and mean age to the growth parameters are substantiallydifferent between a given economy at steady state versus one still in transition.
Inequality emerges in populations with homogeneous populations of individuals of equal ability on landscapesproviding equal opportunity.After a population has achieved a true steady state, the inequality, as measured by the Lorenz Curve[Figure 19(a)], has the following relationship to model parameters. Increasing birth cost [Figure 2], infertility26Figure 3(a)] and puberty [Figure 3(b)], all increase the implied, intrinsic growth rate [Table 4]. As theintrinsic growth rate decreases, mean ages increase [Figures 14 and 15(a)], mean deaths per cycle decrease[Figure 15(b)], total wealth of the population increases [Figures 16(a)], and inequality decreases [Figure 16(b)].These relationships suggest that large inequality at steady state is characteristic of short-lived populationsdriven by high death rates.Increasing birth costs reduce inequality at a much greater rate than increasing infertility or puberty[Figure 16(b)]. The one-to-one mapping of inequality to mean age (and thereby intrinsic growth rate) islost, suggesting an additional effect the rising birth costs have on wealth distributions. And it is perhapscounter-intuitive that these increasing birth costs do not reduce the total wealth of the population thoughthey represent a sunk cost of wealth. These sunk costs may be compensated by the significantly lower deathsper cycle [Equation 5 and Figure 21]. Surprisingly, the high birth cost model economy (f10p1bc40m3), whilethe wealthiest model economy at steady state, is also the most equal model economy.The highest levels of inequalities are found in non-equilibrium periods during and after the initial growthphase of these model economies, when a small number of agents (founders) reproduce into a rich, under-populated landscape. These founders store such significant resources before the population reaches its carrycapacity that even after the carry capacity has been reached, the residual surplus resources decay quite slowly[Figure 22], preserving high inequality and preventing equilibrium for time periods greater than scale of theinitial growth phase. The ability of this simple economic model to simulate these non-equilibrium dynamicsprovides important insights into inequality.
Very simple population-driven subsistence and surplus economies have been defined and then simulated witha spatial, agent-based model. This economic model is made simple by using homogeneous (equal ability)populations operating on equal opportunity (flat) landscapes. Sensitivity studies of model parameters mappedout various dynamic regimes of the population including constant level; and damped, steady, and chaoticmagnitude periodic oscillations; and extinctions. Comparisons of the aggregate dynamics of these simpleeconomies with established population-driven models from mathematical biology and ecology lend credenceto the simple economic model and allowed estimation of implied, inherent rates of growth and delay periods.The extinction events of the simple economic model were examined at the local level and some were shownto occur due to spatial and stochastic characteristics whose details are only available at the individual level.These details reveal the importance of the spatial-temporal structure of populations and its stochastic natureat the individual level. By using additional measures of the distributions of agents’ ages, surpluses, anddeaths the relaxation times to steady state for these economies are shown to be much greater than the time ittakes for the population level to reach actual carry capacity. The driver of these long relaxation times is theoutsized surplus wealth of the founders, the original colonizers of the rich, underpopulated landscape. Theseindividual distributions of surplus resources also provide a direct measure of inequality. The sensitivity ofinequality distributions to model parameters were generated and various regimes were identified. Four modeleconomies were defined for further study and inequality distributions of these model economies were examinedand compared. The dominant effect of founders on the inequality distributions before steady state is achievedis highlighted. Even after the effects of the founders dissipated, significant inequality was shown to exist inpopulations of identical individuals residing in an equal opportunity (flat) landscape. The drivers of theseinequalities, however, are shown to be much different in the steady state phase than during non-equilibriumtransitions. The degree of inequality was shown to be a sensitive to the model parameters with slowerintrinsic growth rates (with lower death rates) producing both lower inequality and greater total wealthof the population at steady state. The inclusion of birth costs made additional contributions to reducedinequality beyond its effects of reducing intrinsic growth while actually increasing the total wealth of thepopulation even though resources were removed at every birth.Simple economies with equal opportunity environments and equally capable individuals generate complexwealth distributions whose inequalities are dependent on the intrinsic growth rate of the population, the costof reproduction, and whether the economy has reached equilibrium.27