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A Crocodilian Population Model using Delay Differential Equations


A. Gallegos, T. Plummer, D. Uminsky, C. Vega, C. Wickman, M. Zawoiski. A mathematical model of a crocodilian population using delay -differential equations. Journal of Mathematical Biology. 57(5), 2008, 737-754.


Crocodilia is a species which exhibits environmental sex determination (ESD), this property is not uncommon for reptiles. The appropriate environmental factor is temperature, so the mechanism is known as temperature-dependent sex determination (TSD). Apart from TSD the species has another interesting characteristic because its life parameters are strongly age dependent.

In this assignment on crocodilian population behaviour TSD and age dependent life parameters play the interesting role of population control. Therefore the main focus of the content and the configuration of the equations is build on these two factors and their effects below.

The impact of temperature influences the sex-determination of crocodiles, because it affects genetic mechanisms and cause the kind of the embryos’ hormonal environment. Temperatures lower than 30°C induce females, temperatures higher than 33°C induce male individuals. The range in between allows both sexes, so it causes no predictable determination.

Differences between breeding temperatures are strongly related with nesting sites. Hence three distinct nesting sites have been observed, each with their own temperature characteristics. In the wet marsh near the water, cool temperature primarily produce female hatchlings. The hot temperatures of the dry levees result in primarily male hatchlings. The dry marsh has an intermediate temperature, resulting in hatchlings of female and males.

File:Nesting areaI.JPG

Fig 1: copied from Gallegos et al. (Journal of Mathematical Biology. 57(5), 2008, 737-754.)

Project Explanation

This project uses the concept of a delay differential equation. In some biological systems, there is a time-delay between when some biological happens (i.e. feeding, procreation) and when that event actually effects the system. This type of mathematical models introduces very interesting dynamics such as oscillations usually depending on the size of the delay. For a more rigorous explanation of delay differential equations, visit Wikipedia: Delay Differential Equations.

Biological models can study a wide variety of systems including molecular, chemical and cellular dynamics. This project focuses on a more ecological system. The research studies gender-specific population levels in crocodiles. Not only does it simulate the whole population, but it focuses on the levels of each gender in the population. Depending on the environment, crocodile eggs will have a different proportion of males to females. This model takes that into account.


It is assumed that femals chose nesting sites by temperature. They prefer sites which are similar to those they were born in. Only if there is a lack of such sites they chose a different class. Thus it is unlikely that female crocodiles lay their eggs in warm nests.

As an effect of this circumstance male hatchlings are much more unlikely than female. Given that the probability of survival is the same for both sexes the population mainly consists of female individuals.

Female crocodiles born in wet marsh (area 1)


File:F1equ juv.JPG


File:F1equ adult.JPG

Female crocodiles born in dry marsh (area 2)


File:F2equ juv.JPG


File:F2equ adult.JPG

Male crocodiles born in dry marsh (area 2)


File:M2equ juv.JPG


File:M2equ adult.JPG

Male crocodiles born in dry leeves (area 3)


File:M3equ juv.JPG


File:M3equ adult.JPG

Description of parameters:

a_i … juvenile removal rates

s …… survival probability

c_i … nesting capacities

tau … delay time

Using Parameters from literature:

k1 = 0.797; k2 = 0.136; k3 = 0.067; dj = 0.22; da = 0.07; b = 0.822;

a1 = dj/b; a2 = da/b; tau = 10*b; c2 = k2/k1; s = exp(-b*a1*tau); c2 = k2/k1; c3 = k3/k1;

For the chosen values of s, tau etc. see matlab code below Media:CrocodiliaDDE.m

Equilibrium points (example)

File:Equ points.JPG

There are similar but longer equation terms for F_2 – m_3 equilibria.

Trivial (=0) and negative equilibrium points are not considered.


The reason to use delay differential equations in this case is the age dependency of the life parameters. We simulated the single delay with the parameter tau equal 8.22 years, which means that the average member of the population becomes sexually mature at the age of eight. Juvenile crocodiles do not reproduce at all and only adults induce population growth.

Changing initial conditions does not alter the behavior of the solution, it just affects the size of the population.

As described above their are much more female than male individuals.

Figure 2 shows the results of our simulation.

File:Result alligators vs timeI.JPG

Fig. 2: Crocodiles vs. time

Other work on delay differential equations

For other studies on the effects of delay differential equations, look into the Effects of Seasonal Growth On A Delayed Prey-Predator Model

Matlab Code


Other Articles

Caravagna, G. Formal Modeling of Biological Systems with Delays. Universita degli Studi di Pisa. Doctorate Thesis. 2009.

In mathematical biology, there are many ways to model a system. Some systems can be studied using a wide array of approaches while other systems are best studied with only one. These differences come from the actual system itself. In this paper, Caravagna uses two different approaches: stochastic and deterministic. The stochastic model is based on the probability that the model will exhibit certain behaviors. This is a usual approach, unless the system itself is chaotic. In this scenario, the model can be difficult to analyze analytically. The deterministic model used is a regular ordinary differential equations. However, these models have there very own difficulties. Caravagna not only studies the uses of each type of approach, she uses examples from other research papers to give examples of her conclusions. Formal Modeling of Biological Systems with Delays

See Also

For more on delay differential equations, see Modelling the Tryptophan Operon, the information on Temporal Limitations: Age Structure in Host-Parasitoid Models, or Optimal Chemotherapy Strategies

Project Categorization

(a) Mathematics Used In this project delay differential equations are used to study the gender specific population distribution of crocodiles. Delay differential equations were chosen because crocodile life parameters are exhibit highly age-dependent life parameters. Crocodiles are considered to become adults when they are sexually mature, which is on average when they are 8.22 years old. The delay time is therefore 8.22 in this project.

(b) Type of Model The model used in this project is a population model. The model assumes that female crocodiles choose nesting sites based upon temperature. It also assumes that no juveniles are able to reproduce and that the survival probability is the same for males and females. The parameters used are juvenile removal rates (a_i), survival probability (s), nesting capacities (c_i), and the delay time (tau).

(c) Biological System Studied The biological system being studied in this project is the population structure of crocodiles. Crocodiles are able to determine the sex of their offspring by the temperature at the location of their nesting sites, and their birth and death rates are strongly dependent upon age. This model takes both of these things into consideration and describes the evolution of a crocodilian population.

Citation of Paper

Rozikov, U.A. "Evolution Operators and Algebras of Sex Linked Inheritance."

This paper was published in January 2013. It is a short overview of recent publications on the topic of discrete-time dynamical systems and evolution algebras of sex linked inheritance. The article goes into some depth about different sex-determination systems, many of which are not yet fully understood. The article discusses the system modeled in this project, where temperature determines the sex of the offspring. It also discusses cases in which adult individuals of a population are able to change their sex mid way through their life. Some examples of hermaphrodites are given, including certain worm and snail species. Sometimes infection can alter sex in species, and for some species we do not understand how sex is determined, such as with the Platypus. The Platypus does not have the SRY gene which is what determines sex for mammals, and no other sex-determining process has been identified.