Group Project #1

Math 4, W02





Aruna Kamath

Tucker Murphy

Samuel Rice

Aisha Siebert

























Oops, Where did the Kakapo Go?


1.* Prior to human habitation of New Zealand, the kakapo population lived relatively unaffected by interspecies competition or predation. From the beginning of their existence till the arrival of humans, between the twelfth and thirteenth century, the kakapo thrived.  With a life span exceeding 60 years (2) and an ever-increasing birthrate they were forced to developed several behaviors as a means of maintaining a population that did not exceed the carrying capacity of the island.  This resulted in the intricate mating ritual now practiced by the species.  Every 3 to 4 years, following the blooming of the perennial rimu trees, kakapo males gathered in lek sites to call the females to mate with a “booming” sound produced by inflating air sacs in their chest and belly (2).  During this mating period each female kakapo had 2 to 3 young, which she reared on her own; and which would be able to mate, themselves after 7 years.  We have chosen to model the kakapo population in equilibrium by a sine function centered around the carrying capacity of 500,000[1], in an attempt to model the variation in population [birth rate: 2.5 young per 2 kakapo; death rate: 1/60 years*total population] over a single breeding period [period: 3.5 years].  In order to construct this model we assumed that the population remained constant[2], excluding changes in number due to abundance or shortage of food, natural disasters, and/or interaction with other species.  However, this model is not entirely accurate considering that the kakapo’s nesting period lasted only 10 weeks (2), and all breeding took place simultaneously within a narrow period of time; therefore leading to a sudden increase in the population followed by a 3 to 4 year period of steady decline as the death rate slowly evened out the population level.  A more accurate function by which one might describe these periodic shifts in population level is the saw-tooth; with the sharp rise representing the sudden increase in population following the mating period, and the longer downward sloping edge symbolizing the slow decline due to a constant death rate. 

Even though the sine function we chose shows an equal period of rise and decline [which would be accurate if the breeding period lasted for 1.75 years, or half the total breeding cycle], it does convey a fairly accurate picture of the swing in population experienced every 3 to 4 years; And, from a mathematical standpoint, as the total population increases the greater magnitude of these shifts is represented in the increasing amplitude of the sine function as the population approaches equilibrium[3].  To make a graph that would retain the sine function at equilibrium, we had to add an additional sine function separately so that the capacitance part of the equation at equilibrium would cancel out the original sine function value. Therefore, by putting in the additional sine function, the overall graph would appear slightly inaccurate. However, if it is assumed that the number of the kakapo born during one breeding cycle at equilibrium is double that of what it was initially the additional sine function makes sense. The difficulty in producing a graph of exponential growth leveling off to equilibrium, similar to what would have been observed in the period from when the kakapo first came into existence till the introduction of human beings into their natural habitat, is that we have no means by which to determine exactly how many kakapo first existed.  Therefore, our graph represents only the five hundred year period, prior to the Polynesians arrival, during which the kakapo population was already in equilibrium.


 Sometime during the 1200’s the first Polynesian settlers came to New Zealand and began hunting the kakapo as well as infringing upon, and destroying parts of the birds’ territories.  Along with the new competition for space, the kakapo population was also affected by the introduction of kiore rats, the first real predators to pose a threat to their numbers. 

Factors that lead to the easy predation of these flightless parrots included: Remaining still and freezing as a natural defense mechanism.  “[One] peculiar habit of the kakapo is to freeze when disturbed, keeping absolutely still and hoping to blend into the background.  Most animals have evolved other forms of defense, such as taking flight, but the kakapo obviously did not need this behavior in the days when New Zealand was free of introduced predators such as rats and dogs [as well as the Polynesian people].”(3) Although this worked in the kakapos’ favor against eagles, which would occasionally target them, it made the kakapos very susceptible to newly introduced mammals who could detect their strong odor over long distances; The tendency of female kakapo to leave their young unattended for long periods after they were ten weeks old; and the tendency of young to make loud noises day and night, making them easily detectable.  In addition to these factors, kakapo did not often associate with others of their species and were relatively defenseless. “Kakapo are solitary creatures.  They gather only to breed, and females are left to raise their chicks alone.  [Each] bird keeps to itself during the day, tucked up in a roost on the ground or in a tree, and ventures out at night to feed.”(3) We determined that although the introduction of two new variables did cause a decline in the kakapo population, they were ultimately heading towards a, somewhat lower, equilibrium but not extinction. Val [1] is defined as the kakapo population[4], Val [2] is the Polynesian population[5], and Val [3] is the kiore rat population[6].  Within each of these factors we included the net effect of the introduced predator; for Polynesians this included direct predation and competition for space, and for kiore rats, competition for space and feeding on kakapo eggs.  The assumptions made in graph 1: that environmental factors remain constant, excluding the two new variables, are applicable to this model.  Initial population values for Polynesians and kiore rats were estimated on the basis of historical context, as well as their net effect upon the kakapo population within the five hundred year time period prior to the arrival of Europeans.  We used the Lotka-Volterra equation to demonstrate the predator-prey interactions between the kakapo, the Polynesians and the kiore rats. This model accurately illustrates the separate interactions of predation by the Polynesians and kiore rats on the kakapo, as well as their interaction with one another.  Final values for each of these populations in the new equilibrium are: kakapo, 230,000; Polynesians, 2,000; and rats, 175,000; indicating that the kakapo suffered a more than 50% decline in population after the Polynesian immigration alone. In this model we removed the sine function for the sake of simplicity; this variation is negligible over such a large period of time.


In 1845, European settlers came to New Zealand (2). Through hunting and deforestation, they further reduced the kakapo population, which by this time had been confined to the central North Island and to the forests of South Island (3). They also introduced several new predators into the ecosystem, namely dogs, cats, stoats, two new species of rats, deer, and possums. To model this graph, we again used the Lotka-Volterra equation to illustrate the additional predator-prey interaction between the human population, the kakapos, and other predators. The assumed external conditions described for graph 1 and 2 hold true for this model.  In this graph, Val [1] represents the kakapo population[7], Val [2] is the net effect of all human inhabitants of New Zealand[8], Val [3] is the total predation factor[9], and finally the deforestation and destruction of the kakapos’ living space is given the constant of [-4/7][10].  As these populations increased, the kakapos were no longer able to maintain their numbers and began to move towards extinction. Without conservation efforts and positive human interference, beginning in 1973, the species would have almost certainly have disappeared; modeled in graph 3 by the kakapo population approaching, and ultimately reaching zero (although in the graph this function dips slightly below zero, indicating an impossible state of negative population), correlating to the exponential growth of both human and predator populations of New Zealand.

Whereas in the initial introduction of predators into the environment (graph 2), a dramatic change in kakapo population was observed, the European arrival accompanied by the introduction of several new predators simply amplified the effects of predation, pushing the kakapo population below the unstable equilibrium point and towards extinction.



2. For several years the kakapo was thought to be extinct until the rediscovery of the species thirty years ago, which spurred the conservation effort. There are currently 62 kakapo remaining in the world. To explain the current state and the future predicament of the kakapo population, we attempted to propose an equation and graphs. This was difficult due to too many variables, notably the positive human intervention and conservation efforts (2). To address this problem we had to form several assumptions namely that the current population falls under the Allee effect. “The Allee effect refers to inverse density dependence at low density” (1). For lex to result in successful breeding, there needs to be a certain amount of male kakapo present. Although originally this mating ritual was developed as a beneficial means of a population control, now that the species is near extinction it is proving to be a handicap. Based on the research we have conducted, this seems to be the primary limiting factor to the Allee effect. Additionally, inbreeding contributes to the Allee effect by increasing morbidity and mortality[11].  The Allee effect is expressed in, “Models of population dynamics in which per capita reproductive success declines at low population numbers predict that populations can have multiple equilibria and may suddenly shift from one equilibrium to another.”(4).  For our graph we chose a fairly simple model of the Allee effect[12] where “the per capita growth rate is negative above the carrying capacity (K) and positive below.  However in the presence of an Allee effect, it also decreases below a given population size, and can even become negative below a critical population threshold (K_).”(1) In our graphs we modeled a stable and unstable equilibrium. Based on our knowledge of the current dynamic of New Zealand, we assumed capacitance to be far below what it originally was before the arrival of humans[13].  Although we were not able to determine the unstable equilibrium (the point below which the kakapo population is independently unsustainable), we estimated it its value to be 75.[14] 

The Allee model produces two equilibriums, a stable and unstable.  All points will tend to move towards the higher, stable equilibrium; except those that fall below the unstable equilibrium, in which case they will approach zero, and the species will become extinct.  The preceding graphs illustrate this phenomenon at our assumed equilibriums.[15]


The migration of human beings to the islands of New Zealand over the past 800 years has greatly affected the native flora and fauna. Polynesians first appeared in the mid 13th Century, introducing kiore rats the effects of which would change the kakapo population forever. Predators were a novelty for the kakapo, against which they could put up little resistance. Before the introduction of these foreign species, indigenous animals like the Kakapo flourished, reaching a stable equilibrium at the carrying capacity of the island. With the introduction of these new predators the kakapo population experienced a major decline.  The kakapo had to adjust to a more complex ecosystem, and reached a new equilibrium at approximately 230,000. We assumed that if the islands had remained at this steady state the kakapo would have continued in the new equilibrium.

            However, the European migration led to an even greater decline in the kakapo population. Along with deforestation (i.e. a loss of breeding grounds) the Europeans brought new, more efficient predators such as dogs, cats, and stoats. A great decrease in population was observed, 230,000 Kakapo were killed in 150 years, and the kakapo population was below 100 individuals by the 1950s, most likely well below its unstable equilibrium. Without human intervention the kakapo population would have probably dropped into extinction.

Now in captivity, kakapos are mating successfully, and with luck they will be able to increase to a level at which they will be independently sustainable. Kakapos are now mating regularly, and their numbers are increasing every year. The kakapos, having such a long life span, should see a recovery as long as humans continue to protect and nurture them.

            Even if the kakapo population is assisted there are still problems these bird may encounter. One such problem is a lack of island growth potential. The kakapo must be placed in a habitat in which the population will not exceed the carrying capacity. Due to the large number of predators on the mainland, the kakapo must be placed on one of the surrounding islands with adequate expansion potential. Another problem the kakapo faces is genetic drift. Since their population has shrunk to such a small number the variation in genes at ever loci will be limited.  Based on the models of the Allee effect, the kakapo population must be above a certain level in order to survive.

* Please note that the population axis of all graphs for question one are scaled by (x*10^3)

[1] This value was chosen arbitrarily as there are no accurate records of kakapo population from this time period, only the generalization that there were “hundreds of thousands”(2) living in New Zealand when the first humans arrived.

[2] A sine curve centered around a single carrying capacity [500,000]

[3] The amplitude of the sine function is proportional to the total population at that point in time.

[4] Initial kakapo population: 500,000

[5] Initial Polynesian population: estimated value of 700 

[6] Initial kiore rat population: estimated value of 1,000

[7] In this case 230,000, the final equilibrium value of kakapo population determined in graph 2.

[8] We have chosen to group European and Polynesian populations into one Val, since both are members of the same species and therefore occupy the same niche within a single ecosystem.

[9] Including direct predation, competition for food/space and consumption of kakapo eggs.

[10]Deforestation assumed to be at a constant rate of 4 acres every 7 years, arbitrarily assigned to fit the negative value needed for this graph.

[11] Inbreeding leads to an increase in the homozygosity of a population. Therefore, there is a greater chance of two recessive deleterious alleles being inherited by a single offspring.

[12] dN/dt=rN(1-(N/K))((N/K_)-1)

[13] 1-(N/5,000)

[14] (N/75)-1

[15] Stable equilibrium: 5,000 Unstable equilibrium: 75