1 A Hare- Simulation Model What happens to the numbers of hares and lynx when the core of the system is like this? Hares O Balance? S H_Births Hares H_Fertility Area KillsPerHead Fertility Births Figure 1: A balancing loop with time delays. Rising hare density causes lynx mortality to fall and hence, when mortality has fallen below fertility, the number of lynx to rise (S, same as hares). When the lynx population and the killing rate per head are large enough, the hare population begins to fall (O, opposite). mortality begins to rise as a consequence and when mortality exceeds fertility, the number of lynx begins to fall (S). This leads to a new cycle. Fertility means (note: here) births that is, surviving babies per head per year. means deaths per head, or the fraction of the population that dies, per year. So they are relative or fractional birth and death rates. See below how the absolute rates (flows) are calculated during a simulation run.
2 Computing the Stocks (levels) H_BIRTHS HARES BIRTHS LYNX Computing the Flows (rates) H_BIRTHS BIRTHS HARES H_Fertility Fertility LYNX LYNX KillsPerHead LYNX Area HARES Figure 2: This is the model of Figure 1. At the starting point the stocks have their initial values; only the rates are calculated at this point. (Symbols with extra corners represent the original red variables.) After the first time step the stocks are updated: the inflow over the past time interval (step) is added to each stock and the outflow is subtracted. The rates are then recalculated because of the updated stocks. The same is repeated after every time step.
3 H_Births Hares 1.25 62500 50000 62500 H_Fertility 1000 Area 50 50 KillsPerHead Fertility 0.25 312.5 312.5 0.25 1250 Births Figure 3: The model is initialized in equilibrium here. What happens to the numbers of hares and lynx? We don t need to run the model because we can see that hare births less deaths (per year) is zero as well as lynx births less deaths. So both net flows are zero at the same time and therefore no values can change given that the area and fertility values are constants. This shows how a perfect balance is theoretically possible in the corresponding real system. (See another initialization on the next page.)
4 H_Births Hares 1.25 50000 40000 50000 H_Fertility 1000 Area 40 40 KillsPerHead Fertility 0.30 375 312.5 0.25 1250 Births Figure 4: Now the user has lowered the initial number of hares by 10,000. So finding food is harder for the predators. The net growth of the hare population is still zero, but the number of lynx is on the fall as the death rate exceeds the birth rate. Once the simulation starts, the declining lynx population draws preying below hare births initiating the rise of the hare population. Since there is space here, let me point out the obvious: the computer does not need the diagram (arrows and variable symbols) for calculating the results; the model equations are quite enough. mortality and kills per head have been defined with a look-up table or graph, also called a graphical function, instead of ordinary equations. See figures 2 and 5 in The and Hare Oscillation Model (Sonoma State University) here: https://asiakas.kotisivukone.com/files/clarity.kotisivukone.com/sonomastateuniversity.pdf Those tables/graphs can also be expressed with the following linear equations: Kills per head = hare density = 1 (hare density / 200 + 0.5) The reason why graphical functions have been used is that they can be easily modified from linear to nonlinear. Besides, the above ordinary equations lack maximum and minimum values; lynx mortality goes even negative (not all right) if hare density becomes higher than 100 in simulation.
5 Simulation Results threshold min: 1134 max: 1374 min: 40 max: 61.5 (hares / ha) 0 2 4 6 8 10 12 14 16 18 20 22 Time in Years Figure 5: The vertical line near the beginning indicates that the lynx population declines while hare density is below the threshold (50 hares / hectare). This lag causes the hare and lynx numbers to oscillate. Whenever hare density is below the threshold, the number of lynx falls whether hare density itself is rising or falling. The same applies to lynx growth: it does not matter whether hare density is rising or falling it simply has to remain above the threshold to cause the lynx numbers to grow. However, lynx growth (decline) begins to slow when hare density reverses course. min: 1134 max: 1374 min: 40 max: 61.5 (hares / ha) 0 1 2 3 4 5 6 7 8 9 10 11 Time in Years Figure 6: A year-by-year representation of the cycle. The curve of the total hare population would be identical with the hare density curve both in shape and time, as opposed to the lynx curve which is shifted in time. The total number of hares varies from 40,000 to 61,500 since the area is 1000 hectares.
6 Hare Density 60 Fertility 55 50 BIRTHS 45 40 60 40 50 50 min: 1160 max: 1282 0 1 2 3 4 5 6 7 8 Time in Years 0.30 0.25 0.25 0.20 (deaths per head) Fertility = 0.25 (births per head) 0 1 2 3 4 5 6 7 8 Time in Years Figure 7: The phase lag between the populations can be illustrated by altering hare density with a slide control. Drag the slide button from 40 to 60 and back again during the simulation run (done already). When does the number of lynx change rapidly? When does it remain unchanged? What is lynx mortality as compared to fertility in those years?
7 Of course, you can t find the smooth and perfectly regular pattern the model produces in any real ecosystem. The model structure is a simplification and so is the simulation output resulting from the structure. Besides, the model is a theory. It is an attempt to explain the dominant feature in the observed historical pattern, in this one in particular. Figure 8: The hare and lynx populations between 1845 and 1935 (almost a century which is why the peaks are very narrow) in Northern Canada as indicated by the numbers of pelts received by the Hudson s Bay Company. The 10-year interval between huge peaks is the most striking feature at least from the systemic perspective. Figure 8 is taken from: Bear, R., & Rintoul, D. 2014. Community Ecology. Connexions. http://cnx.org/content/m46881/latest/ The hare graph happens to be filled with red in figure 8 while in my pictures it is the lynx graph that is shown in red. Also note that the numbers on the y-axis actually refer to the hare population, not the lynx one. You can verify it by looking at the corresponding figure in this introduction to population growth: http://www.nature.com/scitable/knowledge/library/an-introduction-to-population-growth- 84225544 (Continued on the next page)
8 Further Analysis and Experimentation This section provides some more of the same plus supplementary information though I am not able to give the latter very much. Recall the simulation we had in which the lynx population was launched from 1250 and the hare population from 40,000. Let s call it the standard simulation. The first cycle, or any cycle for that matter, for both populations is depicted in figure 9. So hare density starts from 40 and the number of lynx from 1250 (then falling to 1134). 1374 1134 threshold Hare 50,000 or hare density 50 Hare Figure 9: As the dots indicate, the entire growth of the lynx population occurs while hare density is above the threshold. In the middle of its growth, right above its initial level, the lynx population is large enough to drive the hare population over the edge, then pressing its declining prey even harder until the second dot, after which the pressure is gradually relieved. See that line, the threshold, in figure 9. Imagine that it represents lynx fertility. Where is then lynx mortality while hare density is arching above the line of 50? The answer is simple when you think about it: First of all, lynx mortality must be below the line since the number of lynx is growing. Then comes everything else, like the idea that lynx mortality is very low when hare density is very high. Indeed, the logic is straightforward in this model. mortality forms a U-shaped mirror image of hare density starting from and coming back to the fertility line at which time there is no change in the number of lynx, whether it is their highest or lowest number. So in regard to lynx, we might call all the four dots here zero dots as an aide-mémoire. Invent a better one! The change rate is zero whenever there is no change in some level a tautology but in this model, at the dots, it means that the lynx death rate equals the birth rate (meaning mortality = fertility) causing only the net rate of change to be zero. At the same time, by contrast, the hare population is rising/falling fastest due to the position of lynx. The change rate is far from zero, but that s obvious. This must be an easy question then (besides, answered earlier): When does the lynx population grow fastest?
9 That s right: in the middle of its growth due to highest hare density and, correspondingly, lowest lynx mortality. The growth rate first increases from zero and then decreases to zero, which makes the growth of the lynx population slightly S- shaped. In passing, it should be noted that the exact time of fastest growth depends on whether we look at the relative (fertility less mortality) or absolute (births less deaths) growth rate. The latter depends on the number of lynx in addition to hare density (through lynx mortality), and so the peak rate is a little later than the peak relative rate, which is highest exactly when hare density is highest. The number of lynx begins to fall after the second dot. Why? How would you explain it in terms of lynx mortality given that mortality is back at the fertility line when the lynx population is at the top? The answer comes here: As hare density falls below the threshold, lynx mortality correspondingly rises above the fertility line. That s it. What follows is an inverted U shape representing lynx mortality arching above the fertility line while hare density is plotted as the opposite (normal) U shape below the threshold. Here, too, the gap between lynx mortality and fertility is largest in the middle of the fall of the lynx population; the decline is at its fastest. Before shifting from growth to decline, however, the lynx population enters a phase where it still grows while its prey is already going downhill. (See figure 9 above.) How convincing is the idea that the predator population lags behind this way? If lynx mortality jumped right over to the other side of the fertility line when hare density begins to fall, the delay would disappear. Could it not at least move back to the line a little faster? But then again, why did mortality veer so far off the line in the first place? And what is far? A relative matter. I am not the right person to deal with this, obviously, but I think natural growth, like that of lynx, must slow before it stops entirely. In any case, it is what happens here during the delay in reversing course. (I ll return to that subject later if I can find a better answer.) We have now dealt with lynx mortality defined here as deaths per lynx (per year), but the model also contains implicit hare mortality, which can easily be made explicit by adding it to the model as a variable: deaths per hare. H_BIRTHS Hares H_ H_Fertility Figure 10: Hare mortality equals the preying rate / hares.
10 Included this way (figure 10) hare mortality does not influence other variables in the model but can be plotted in simulations and then compared to hare fertility, which is a constant. This is done below, where a cycle is presented again with aid of dots and auxiliary lines. The constant hare fertility equals 1.25 as illustrated by figure 11. 1374 1250 Hare 50,000 or hare Hare density 50 Hare falling when mortality on this side H. mortality H. fertility 1.25 40,000 or 40 Figure 11: Note! The number of lynx and hare mortality change in the same direction, as opposed to the inverse or negative relationship between hare density and lynx mortality dealt with previously. This simply means that a large lynx population is bad for hares. All the sections between two dots timely correspond to each other. Since hare mortality is directly proportional to the number of lynx, why not make it dependent on lynx alone in the model? Interestingly, the initial value of lynx is 1250 while that of hare mortality is 1.25. Do you see the connection? Now we could simplify the model structure by maybe dividing the number of lynx with the area (1000), thereby getting hare mortality, and then redefining hare deaths (preying) as hares * hare mortality just like lynx deaths equal lynx * lynx mortality. What a dim bulb I have been all this time! (The lesson: Do your math homework. You never know what you ll end up doing as a grown-up.) But we will return to this later. The hare population grows when the number of lynx is below 1250. This is an arbitrary line in the standard simulation, like the threshold for the lynx population to grow, but while hare density surpassing 50 is good for lynx, the lynx population going above 1250 is bad for hares. However, to avoid confusion I do not refer to 1250 as a threshold. To summarize, here is another aide-mémoire: A large lynx population is bad for hares hare falls; a small hare population is bad for lynx lynx falls. To be more precise, or looking from a different angle, it gets gradually worse for the hare population already when the number of lynx begins to grow from the bottom level since hare mortality begins to rise, too, without delay. And when the number of lynx begins to fall, it gets gradually better for the prey, whose fall begins to slow at once. The same goes for the lynx population but the other way round.
11 (Unfinished yet)