How-To: Stage-Structured Matrix Models

Happy Thanksgiving everyone!

So this week I constructed several stage-structured matrix models- aka Lefkovitch models- to estimate the finite and intrinsic rate of increase of the weevils Neochetina eichhorniae and N. bruchi under laboratory simulated Fall and Winter conditions in the Sacramento-San Joaquin River Delta. This work is in conjunction with a postdoctoral researcher- Angelica Reddy and Paul Pratt’s laboratory at the USDA to test the temperature performance of these biological control agents.

As you know, if insects are adapted to warm weather- they don’t perform very well in colder temperatures and this can be very applicable to biological control agents (such as the two Neochetina weevil species) that are brought from their tropical origins to colder regions to control invasive species.  Below is a cute cartoon I had a scientific illustrator Jacki Whisenant draw for me, and for a new children’s book we are writing…stay tuned!

Copyright 2017, illustration by Jacki Whisenant and made for Julie Hopper

Because we are working in the laboratory on these two species we are able to gather a lot of life history parameters of the weevils undergoing Fall and Winter conditions. These parameters include: development time and survivorship of the different insect stages (egg, III instars of larva, pupa, pre-reproductive adult and reproductive adult), as well as the emerging sex-ratios, and longevity and daily and lifetime fecundity of the reproductive female adults.

From these parameters we can conduct several different analyses to approximate the finite rate of increase, intrinsic rate of increase, generation time, doubling time and net reproductive rate of a species to understand more about their potential population growth rates (which of course is important for biological control).

My favorite way to approximate these population growth parameters for insects is to use a stage-structured matrix model (Lefkovitch model). There are other methods you can use as well- but I won’t go into that here. If you would like to read more see the citations at the end of this blog.

Instead, I will provide a how-to tutorial since while I was working on these matrix models as a graduate student- I realized there is a lack of tutorials on the web on how to construct these models in an intuitive manner. I got lucky  as both my PhD adviser and one of my lab mates (whom had already done the research on stage-structured matrix models) helped me understand how to construct and interpret the models. In the name of paying it forward- I am attaching here an excel worksheet that has all of the calculations and formulas that demonstrate how to construct these stage-structured matrix models (see link). 


In my next blog- I will detail how to use this resulting matrix and input it into the package popbio (Stubben and Milligan 2007) for calculation of finite rate of increase (lambda), intrinsic rate of increase (r), doubling time, generation time, net reproductive rate and much more!

Disclaimer- this is for insect stage-structured matrix models only as calculations differ for plants and vertebrates typically.

Here are some of the calculations that are built into the excel formulas: 

Screenshot 2017-11-22 12.53.45

Below is another screenshot of the file: 

Screenshot 2017-11-22 12.57.42

Below is a diagram from the famous study on Loggerhead sea turtles that explains the flow of this matrix better. However be aware that the matrix above and in the attached excel sheet-calculates gamma as 1/duration which is very different than the famous example on turtles (below), and from any matrix with plants- mainly due to life history differences among plant, invertebrates and vertebrates.

Screenshot 2017-11-22 13.10.04.png
A Stage-Based Population Model for Loggerhead Sea Turtles and Implications for Conservation Author(s): Deborah T. Crouse, Larry B. Crowder, Hal Caswell Source: Ecology, Vol. 68, No. 5, (Oct., 1987), pp. 1412-1423


Caswell H (2001) Matrix Population Models: Construction, Analysis, and Interpretation. Sinauer Associates, Sunderland


Awesome powerpoint by Chris Free at Rutgers 

Another awesome tutorial on primarily the Leslie matrix from UCSC



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