# Introduction

PADRINO is pretty thoroughly checked in terms of reproducing published IPM behavior, but it is not strictly checked for the mathematical behavior of each IPM. Therefore, there are times where using IPMs from PADRINO in ways that the original authors did not use them may yield results that are either bizarre or biologically impossible. The purpose of this vignette is to show how to handle these so that these IPMs are still usable.

# Fundamental operators

The most common issue we’ve found is when a function that predicts survival probabilities equals 1 for some range of trait values in the IPM. This doesn’t cause any mathematical issues in many analyses, and so often goes unnoticed by authors publishing their IPMs (the author of this document included - example to follow shortly!). However, to our knowledge, no species becomes immortal regardless of their size, and so this will cause issues for any longevity-related questions. Additionally, a mathematical issue arises whenever one needs to compute the fundamental operator for a model with this.

## Overview of the problem

The fundamental operator, $$N$$, can be thought of as the amount of time an individual will spend in state $$z'$$ given state $$z$$ before death. It is given by

$N = (I + P + P^2 + P^3 + ...) = (I - P)^{-1}$

This Neumann series corresponds to the geometric series $$1 + r + r^2 + r^3 + ... = (1-r)^{-1}$$ and is valid for any real number $$|r| < 1$$, and remains valid provided survival probabilities for individuals are less than 1. Clearly, this computation is no longer valid when an IPM’s survival model predicts survival probabilities that are equal to 1.

## Diagnosis

This most often manifests as negative values in the fundamental operator, which propagate through the rest of the analysis and return nonsense results. Below is a quick example of this manifesting in an IPM for Lonicera maackii when computing mean lifespan ($$\bar\eta(z_0) = eN$$).

library(Rpadrino)

data(pdb)

problem_ipm <- pdb_make_proto_ipm(pdb, "aaa341") %>%
pdb_make_ipm()

P <- problem_ipm$aaa341$sub_kernels$P N <- solve(diag(nrow(P)) - P) range(colSums(N)) ## [1] -348825.70 -30555.21 According to this, Lonicera maackii is expected to live between negative 348,800 and negative 30,555 years. This seems incorrect. Therefore, we should inspect what’s going on with the $$P$$ kernel, and more importantly, the functions that comprise it. We’ll use a couple functions from Rpadrino for that: kernel_formulae(problem_ipm) ##$aaa341
## P: s * g
## F: Ep * Fp * Fs * Fd
vital_rate_exprs(problem_ipm)
## $aaa341 ## s: 1/(1 + exp(-(si + ss1 * size_1 + ss2 * size_1^2))) ## g_mean: gi + gs * size_1 ## g: dnorm(size_2, g_mean, g_sd) ## Fp: 1/(1 + exp(-(fpi + fps * size_1))) ## Fs: exp(fi + fs * size_1) ## Fd: dnorm(size_2, fd_mean, fd_sd) We can see that P = s * g, and since g is given by a probability density function, we are unlikely to find much going on there. Thus, we want to inspect the values for s. How do we do that? By default, ipmr, the engine that powers Rpadrino, does not return individual function values in IPM objects. Therefore, we’ll need to rebuild the IPM, and tell ipmr to give us those by setting return_all_envs = TRUE. Then we can ask for the vital rate functions using vital_rate_funs(). problem_ipm <- pdb_make_proto_ipm(pdb, "aaa341") %>% pdb_make_ipm(addl_args = list(aaa341 = list(return_all_envs = TRUE))) vr_funs <- vital_rate_funs(problem_ipm) vr_funs ##$aaa341
## $aaa341$P
## s (not yet discretized): A 500 x 500 kernel with minimum value: 0.1063 and maximum value: 1
## g_mean (not yet discretized): A 500 x 500 kernel with minimum value: 18.6208 and maximum value: 497.9516
## g (not yet discretized): A 500 x 500 kernel with minimum value: 0 and maximum value: 0.0337
##
## $aaa341$F
## Fp (not yet discretized): A 500 x 500 kernel with minimum value: 0 and maximum value: 1
## Fs (not yet discretized): A 500 x 500 kernel with minimum value: 30.2437 and maximum value: 4705.0552
## Fd (not yet discretized): A 500 x 500 kernel with minimum value: 0 and maximum value: 0.3308

Ah ha! The maximum value of s is 1! This is problematic for us. How do we fix this?

## Fixing the problem

The quickest way is to modify the survival function to be a parallel minimum of the original survival function (i.e. the one that the authors used) and some maximum survival value that we choose ourselves. For the purposes of this example, we’ll use 0.98 as the maximum survival probability. Rpadrino has two functions to help with this: vital_rate_exprs<- and pdb_new_fun_form(). These are used together to insert a new functional form into a proto_ipm so that we can make changes to the IPM without having to think too much about how a proto_ipm is actually structured.

problem_proto <- pdb_make_proto_ipm(pdb, "aaa341")

vital_rate_exprs(problem_proto) <- pdb_new_fun_form(
list(
aaa341 = list(
s = pmin(0.98, plogis(si + ss1 * size_1 + ss2 * size_1 ^ 2))
)
)
)

good_ipm <- pdb_make_ipm(problem_proto,
addl_args = list(aaa341 = list(return_all_envs = TRUE)))

vital_rate_funs(good_ipm)
## $aaa341 ##$aaa341$P ## s (not yet discretized): A 500 x 500 kernel with minimum value: 0.1063 and maximum value: 0.98 ## g_mean (not yet discretized): A 500 x 500 kernel with minimum value: 18.6208 and maximum value: 497.9516 ## g (not yet discretized): A 500 x 500 kernel with minimum value: 0 and maximum value: 0.0337 ## ##$aaa341$F ## Fp (not yet discretized): A 500 x 500 kernel with minimum value: 0 and maximum value: 1 ## Fs (not yet discretized): A 500 x 500 kernel with minimum value: 30.2437 and maximum value: 4705.0552 ## Fd (not yet discretized): A 500 x 500 kernel with minimum value: 0 and maximum value: 0.3308 P <- good_ipm$aaa341$sub_kernels$P
N <- solve(diag(nrow(P)) - P)

range(colSums(N))
## [1]  5.462124 50.007606

These values are far more reasonable!