Interspecific Competition Between Species Using Logistic Growth and Discrete-time Dynamical Systems

The Logistic Growth Law: How is it related to predator and prey models?

    What is a discrete-time dynamical system?

          –>  a dynamical system whose state evolves over state space in discrete time steps according to a fixed rule (in this case, constant K)

 N1(t+1) = rN1(t)(1- (N1(t)+N2)/K) + N1(t)

    This system shows the relationship of the two animals over time in relation to time (time steps). Although the model is measuring the differences in population between two different species, only one species needs to be actively monitored to gather the data regarding competition. 

    N1(t+1) = the births of population one in regards to population two over time 

    N1 = the births of population 1 

    N2 = the births of population 2

    r = (b-d)[birth - death] <- growth rate  

    K = the carrying capacity (the amount that when exceeded, will result in a decrease in population) 

• Carrying capacity r, (K) and N2 are constants in this model 

This graph exhibits the populations of two competing species over time when put into the same environment [image credit: "Ecological and Evolutionary Consequences of Species Interactions", by Sinauer Associates Inc, Principles of Life]

 

    What is carrying capacity(K)? In ecology the carrying capacity is the total amount of individuals allowed for a population, like a threshold. If the population exceeds this total amount, then there will start to be a decline/decay seen in the number of individuals, which can lead to the extinction of the species and the overpopulation of another. This decline occurs because the equilibria of coexistence is thrown off. The overpopulation will diminish resources and start a phenomena in species two that will result in a 'lag', further leading to population decline.

These isoclines show the thresholds of which the two species must not cross to maintain equilibria. [image credit: Relyea, R. (2020), "Ecology: The Economy of Nature", Ch. 15]

    How does this model apply to real life competition between populations? 

Interspecific Competition Between Corals and Sea Sponges

Sponges and corals are very reliant each other, because they share the same environment and resources. If the sponges become too abundant they take up too many of the resources that the corals need. This can result in death of coral reefs and coral bleaching. Once the corals begin to die, the sponges will begin to die too, shortly after. The increase of global water temperatures also decreases the algae growth on corals, resulting in coral death. 

How to determine the equilibria stability:

Using differential equations, and slope to find the equilibria 

How will this look over time?

    The image above shows the work to determine the equilibria points between two populations that are competing in the same environment. This takes into account nothing else except for the competition between the two species, incorporating growth rate and carrying capacity.

            - At (0,0) the populations are stable, since there are zero individuals

            - At (40,580) the populations are stable. They will not decline or increase after 580 individuals. N1 (species 1) must be at 40 individuals and N2 (species 2) must be at 580 individuals.

            - At (112.5, 0), species x, (N1) as in species 1, is stable when species two is at zero.

            - At (0,700), species y, (N2), species 2, is stable when N1 is at zero.

    Half carrying capacity is the optimal number of individuals for there to be sustainable growth and competition between species. After breaching 580 individuals, overpopulation of species 2 (N2) will occur, which leads to the decline of N1 over time. If N1 is below 40 at any point when in the presence of N2, extinction will occur. 

    There are wide parameters for this relationship to be successful. But with the equilibria points clear, it is easier to test the constants and the variables. What happens when N1 is changed? What about when K, r, and N2 are changed? Below is an example of work done when changing all three of the constants. How does this change the results? 

    After testing different numbers with this model, it was determined that changing the value of "r" changes the rate of N2. Changing the rate of r can also affect the K (carrying capacity). Increasing "r" decreases the species 2 constant. Changing N2 every step will throw off the equilibrium in small increments, gradually increasing or gradually decreasing. To achieve equilibrium, K must be at 100, r must be at approximately 1.7-2, and N2 must be between 3 and 7. N1 has more flexibility, but cannot go over 1000. 

    If any of these limits are breached, extinction will be the end result of both species. Even though N1 and N2 rely on each other, they must stay within the parameters to ensure sustainable growth and longevity in their shared ecosystem. 

    This hypothetical growth rate and overpopulation prediction results in species one overtaking species two in a short amount of time, which results in decline of species two, then species one. As climate continues to heat the waters, the sponge populations far exceed that of the corals, resulting in high coral deaths. This is an ecological butterfly effect, leading to the death of sponges and other reef life in the ocean. If this situation was to one day become true, it becomes a permanent threat to the longevity of ocean habitats and ocean life, as well as water quality. 

Example of the dramatic decrease in coral-algal interactions as abiotic factors (light, weather, etc) worsen. [image credit, Brown et al. (2018), "The Dynamics of Coral-Algal Interactions in Space and Time on the Southern Great Barrier Reef"] 

    Using this model allows scientists to predict the trends of population growth and decline over time between predator and prey species, as well as between mutualistic species relationships. As seen in the example above, the growth rates and carrying capacities are kept at a constant, the only changes being initial populations and growth rates of population N1 over time. This equation is used to find the equilibria for species one, and can even be used for species two.

    This is a very important model that must be constantly observed across a variety of competitor species, especially as the climate across the globe drastically changes. The relationships between these animals change when the ecosystem their inhabit changes. This occurrence will eventually lead to the extinction of many animal and plant species, resulting in the death of numerous ecosystems between and within the global hemispheres. Other things such as change in water supply and precipitation levels can be affected by this change in animal populations. 

     If interested, the video above provides an explanatory process of calculating the equilibria of species one and species two through both logistic growth models and discrete-time dynamical systems.